I go by MiTerSkyArk. As of 10/30/2024 I am fairly new to googology, having done it for only about half a year, but being interested in large numbers since 8th grade. However, my father introduced me to the idea of a googol, and even a googolplex, when I was just ~6 years old. Everything I made that uses Brick Notation came about when me and my friends had a competition to see who could make the strongest function within a week. I got the idea for my function and notation (along with the whole idea for the contest) by staring at a brick wall at school, thinking about how I could use something similar in googology. The course function is named so, because a line of bricks is called a course, and the function generates long lines of arrays. Ever since May 2024, me and my friends have been making a gigantic salad number, using everything we can find, along with what we make.
I am trying to make a very powerful notation. This process takes much trial and error, however. Hence, My many notations.
Big numbers are fun and all, but y'know what's more fun than a big number? An even bigger number.
Brick notation is the notation I developed when me and two other friends decided to have a competition to see who could create the stronger function. My winning function was built upon this notation. The notation was made fairly complicated, in order to assure my triumph. It is also built upon other rules/notations, such as:
a!b = (...((a!)!)!)...! with b factorials
^ = ↑
a![b] = a^b(a-1)^b(a-2)......^b2^b1 with ^b meaning b number of arrows.
*The above declarations are assumed for all of my work*
The notation itself is defined as so:
<a> = a!a Example: <4> = 4!4 = (((4!)!)!)!
<a, b> = a![b!] Example: <4, 3> = 4![3!] = 4![6] = 4^^^^^^3^^^^^^2^^^^^^1
<a, b, c> = a![b![c!]] Example: <4, 3, 5> = 4![3![5!]]
<[a]> = <a, a, a...a, a> with a number of a's.
An array can have any number of entries, each one expanding by a lot. But there are some rules about the arrays:
In an array, the first entry must be equal to or greater than 3, as 2 will just result in the whole thing just equaling 2.
Only the final entry is allowed to be 2 or 1, as they are still able to effect the second to last entry. If there are 2's elsewhere, they are ignored.
All 1's are ignored (E.g.: <3, 5, 1, 5, 1> = <3, 5, 5> E.g.2: <5, 4, 3, 2, 1> = <5, 4, 3, 2>)
Also, arrays can be stacked. it can look like:
<n> \n/
<a> or <a> or <a>/<n>
Of course, these arrays can have any number of entries like so: <a, b>/<n, v> or <a, b, c>/<n, v, w>
An array on top of another array means that whatever the top array equals to, the array below it is repeated that many times. Example:
\3, 4/
<4, 3> = <4, 3, 4, 3...4, 3> with <3, 4> repititions of 4, 3. They can also be stacked to any height. A square bracket array before the main array also entails something.
[t]<a, b, c...> = <<...<<a, b, c...>>...>> with t sets of array, meaning a, b, c... is put through an array t times. It is also possible to put curly braces after the amin array to expand it.
<a, b, c>{2, f}=<<a, b, c>, <a, b, c>....<a, b, c>> with f <a, b, c>'s in the array.
<a, b, c>{3, f}=<<a, b, c>{2, f}, <a, b, c>{2, f}....<a, b, c>{2, f}> with f <a, b, c>{2, f}'s.
<a, b, c>{4, f}=<<a, b, c>{3, f}, <a, b, c>{3, f}....<a, b, c>{3, f}>> with f <a, b, c>{3, f}'s.
With the first number (h) being able to be any number equal to or greater than 2, and the second number can be any number. These "extensions" can be used together like so:
[t]<a, b, c...>{h, f} and can be stacked like so: [t]<a, b, c...>{h, f}/[t]<a, b, c...>{h, f}/...[t]<a, b, c...>{h, f}
<333>/<333> = Tritron
<333>/<333>/<333> = Centra-Tritron
<[10]> = Dec-Little
<[100]> = Centiny
<[1,000,000]> = Mega-Little
<[<[<[<[<[5]>]>]>]>]> = Quin-Little
[100]<[<[<[<[<[100]>]>]>]>]> = Cent-Big
<[<[<[<[<[1000]>]>]>]>]>{1000, 1000} = Kilo-Big
<[<[<[<[<[1000]>]>]>]>]>{1000, 1000} /<[<[<[<[<[1000]>]>]>]>]>{1000, 1000} /<[<[<[<[<[1000]>]>]>]>]>{1000, 1000} = Kilo-Huge
<3> = 3-Little
<4> = 4-Little
<[3]> = 3-3-Little
<100, 100>{100, 100} = Gwaloogia
ab = <a, a, a...a, a>/<a, a, a...a, a>/...<a, a, a...a, a> with b entries of a in each stack, and b number of stacks. These are mainly used while inside arrays, like so:
<ab, cd...>
Putting a # before the number in the subscript implies it's usage for every entry after the one it is used on. Example:
<a#b, c ,d...>=<ab, cb, db...>
Putting two numbers before the # also implies something. Example: <a2, 1#10, b, c>=<a10, b12, c14> It adds 2 to the subscript, since the first number is 2, the second number is 1 (meaning the fist operation) and it starts at 10 because of the 10 after the #. Another example: <a4, 2#3, b, c>=<a3, b12, c52> It starts at 3 because pf the 3 after the #, it gets multiplied because of the 2, and it's x4 because of the 4.
<333333, 4#333, 333, 333>/<333> = Kilo-Tritron
<333333, 4#333, 333>/<333, 333>/<333, 333> = Mega-Triton
<[100#100]> = Cent-Little
<[1,000,00010, 2#10]> = Mega-Medium
<[<[<[<[<[5#5]>]>]>]>]> = Quin-Medium
<[<[1,000,000,0001,000,000,000, 1,000,000,000#1,000,000,000]>]> = Giga-Huge
There are bigger subscipts however. For example, <[ab, c#d, e#f]> the "b" and "c" effect "e" in the same way "d" and "e" effect "f", which vastly effects the subscript's growth throughout the array. There can be any amount of variables and # in the subscripts, like
<[ab, c#d, e#f, g#h, i, j#k]> or <[ab, c#d, e#f, g#h, i, j#k, l#m]> each one effecting the next in extremley vast ways.
For example <[52, 2#2, 2#2]> = <<[52, 2#2]>, <[52, 4#2]>, <[52, 8#2]>, <[52, 16#2]>, <[52, 32#2]>>
Course(a)=<a, (a-1), (a-2).....3, 2, 1>
Course(a, b)=<Courseb(a), Courseb(a-1), Courseb(a-2)...Courseb(3), Courseb(2), Courseb(1)> with Courseb(a) meaning Course is applied to a, b number of times, and when used with multiple inputs, it is like so: Course2(b, c)=Course((Course(b, c))b, (Course(b, c))c)
Course(a, b, c)=<Coursec(a, b), Coursec(a-1, b), Coursec(a-2, b)......Coursec(2, b), Coursec(1, b), Coursec(a, b-1), Coursec(a-1, b-1)...Course(1, 1)>
with Coursec(a, b) = Coursec-1((Course(a, b))a, (Course(a, b))b)
Basically, the first entry in the array is the same as the original function, minus the last entry, which is instead used to apply the function multiple times. The next entry's first entry is reduced by one, and repeats this until it itself is one, at which point, it rolls back, and the second entry is reduced by one. The first entry is then reduced again and again until it reaches one again, at which point the second entry is reduced by one again. This same process happens to any amount of entries. Once one entry reaches one, the next time it's rolled back and the next entry is reduced by one, until every entry is 1.
Course([a])=Course(a, a, a...a, a) with a a's.
Course(4) = Tetrartet
Course(5) = Quiniq
Course(10) = Decaced
Course(100) = Centatnec
Course(1,000) = Kilolik
Course(1,000,000) = Megagem
Course(Course(100)) = Mega-cent
Course(3, 3) = Duotri
Course(3, 3, 3) = Triotri
Course(4, 4) = Duotetra
Course(4, 4, 4) = Tritetra
Course(Course(3, 3, 3)) = Mega-Triotri
Course(100, 100) = Duocent
Course(100, 100, 100) = Tricent
Course(5, 4, 3, 2, 1) = Prev-quin
Course(10, 9, 8, 7, 6, 5, 4, 3, 2, 1) = Prev-dec
Course(<10, 10, 10>, <10, 10, 10>, <10, 10, 10>) = Mega-dec
Course([10]) = Decadec
Course([100]) = Centacent
Get the result of Course([10,000,000]) and put it into a variable called X(1).
Create X(2) and set it to the result of Course([X(1)])
Create X(3) and set it to the result of Course([X(2)])
Repeat this process until you reach X(X(X(X(X(100)))))
This number I call Macro-Course.
n#1! takes all of the digits from n to 1, and concatenates them together, also n#!1 is the same as n#!. Examples:
4#! = 4,321
10#! = 1,098,765,321
15#! = 151,413,121,110,987,654,321
The next step is n#2! which takes n#1!, and applies #1! to it again. Examples:
4#2! = 4,321#1!
10#2! = 1,098,765,321#1!
Naturally, the next step is n#3! which takes n#2! and applies #2! to it again, takes that number, and applies #2! again. Examples:
4#3! = ((4#2!)#2!)#2!
10#3! = ((10#2!)#2!)#2! = 10#2!3
n#k! = n#(k-1)!k
Of course, k can be any number.
n*1! = n*! = n#1! = n#!
n*2! = ((...(n*!)*!)...)*!)*! with n *!
n*3! = ((...(n*2!)*2!)...)*2!)*2! n times
n*k! = ((...(n*(k-1)!)*(k-1)!)...*(k-1)!)*(k-1)! n times
This is a very strong "extension" of the Course function. It starts as so:
Quorvask(1) = Course([<[10↑↑↑10]>])
Quorvask(2) = Course([<[Quorvask(1)]>])
Quorvask(3) = Course([<[Quorvask(2)]>])
Quorvask(n) = Course([<[Quorvask(n-1)]>])
Quorvask(10) = Quordec
Quorvask(50) = Quorquindec
Quorvask(100) = Quorcent
Quorvask(1,000) = Quorkil
Quorvask100(10) = Quorcendec
Quorvask100(1,000,000) = Quorcenmega
A subscript variable (k) after the function name implies Quorvask(1)'s value is Course([<[k]>]) rather than Course([<[10↑↑↑10]>]).
Quorvask10↑↑↑↑10(100) = Megacendec
Quorvask10↑↑↑↑10100(100) = Megadoublecendec
Quorvask10↑↑↑↑↑↑10(100) = Gigacendec
QuorvaskG(64)(1,000) = Kilograham
QuorvaskTREE(3)(1,000,000) = Treemega
Short for MiTerSkyArk factorial array notation. I made this because hyperfactorial array notation makes no sense to me, but I wanted something more powerful than what I already was using.
n![m] = n^m(n-1)^m(n-2)^m(n-3)...^m2^m1
n![m, a] = n![m![m![m![m!......m![m]...]]]] with a m!'s. Example: 3![4, 2] = 3![4![4![4]]] The 4 is repeated on itself 2 times.
n![m, a, b] = n![m![m![m![...[m, a]], a![...[m, a]]], a![m![...[m, a]], a![...[m, a]]]]......a![m![m![[...[m, a]], a![[...[m, a]]], a![m![[...[m, a]], a![[...[m, a]]]]] with b recursions. Example: 4![3, 5, 2] = 4![3![3![3, 5], 5![3, 5]], 5![3![3, 5], 5![3, 5]]] The most basic array within this array is 3![3, 5] and 5![3, 5], and these are used by 3 and 5. Essentially, the two first entries get their own arrays of themselves, the third entry amount of times.
Example of four entries: 3![3, 3, 3, 3] = 3![3![3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]]], 3![3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]]], 3![3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]], 3![3![3, 3, 3], 3![3, 3, 3], 3![3, 3, 3]]]]
and so on. An array can have any number of inputs.
n![{m}] = n![m, m, m, m...m] with m m's.
n![{{m}}] = n![{m}, {m}, {m}...{m}, {m}] with m {m}'s, with each {m} being m, m, m, m...m, m with m m's.
n![{{{m}}}] = n![{{m}}, {{m}}, {{m}}...{{m}}, {{m}}] with m {{m}}'s, with each {{m}} being {m}, {m}, {m}, {m}...{m}, {m} with m {m}'s.
and so on. There can be any number of {}'s. n![{{...{{m}}...}] with a braces = and array of {{...{{m}}...}} with a-1 braces, that is m long, with the whole thing simplifying to n![m, m, m...m, m] with m^a m's.
n![{m, a}] = n![m, a, m, a, m, a...m, a, m, a] with m![a] recursions of m, a.
n![{{m, a}}] = n![{m, a}, {m, a}, {m, a}, {m, a}...{m, a}, {m, a}] with m![a] recursions of {m, a}.
n![{m, a, b}] = n![m, a, b, m, a, b...m, a, b, m, a, b] with m![a, b] recursions of m, a, b.
n![{{m, a, b}}] = n![{m, a, b}, {m, a, b}, {m, a, b}...{m, a, b}] with m![a, b] recursions of {m, a, b}.
There can be any number of entries in the array, and any number of {} around them.
n![ma] = n![{{{...{{{m}}}...}}}] with a {}'s.
n![{ma}] = n![ma+1]
n![ma, a] = n![{{{...{{{m]]]...}}}, a] with a {}'s around the m.
n![ma, bc] = n![{{{...{{{m}}}]...}}}, {{{...{{{b}}}...}}}] with a {}'s around the m, and c {}'s around the b.
Any number of entries can have subscripts.
n![m#a, b, c, d...] applies the "a" subscript to every entry after it.
It also works with n![{m#a}], which rather than just adding 1 to a, repeats ma, m times.
n![m, a, b...]x means the factorial array is applied x times to n. Example: n![a, b]3=((n![a, b])![a, b])![a, b]
10![100] = Googolfan
10![100, 10] = Googolfanplex
50![10, 9, 8, 7, 6, 5, 4, 3, 2, 1] = Halcendecadim - The first half of the name, "halcen", comes from half-cent as the main number is a 50, which is half of 100, which the prefix "cent" is typically used to represent 100. The second half, "decadim", comes from deca, meaning 10, and dim, short for diminish. This is because the array consists of 10, 9, 8...2, 1.
3![3, 3, 3] = Triple-Trial
3![3, 3, 3, 3] = Quadruple-Trial
4![4, 4, 4, 4] = Quadruple-Quadral
10![{10}] = Squardeca
1,000,000,000![{{10}}] = Gigamasdec - The prefix, "giga", means one billion, because of the one billion on the front of the number. The "mas" is short for "massive." The "dec" means 10, because of the 10 in the array.
10![{10#10}] = Cudecabig
100![{100#1000}] = Yottamacent
10![{100}] = Kilogoogolfan
10![{{100}}] = Megagoogolfan
10![{{{100}}}] = Gigagoogolfan
10![1004] = Teragoogolfan
10![1005] = Petagoogolfan
10![1006] = Exagoogolfan
10![1007] = Zettagoogolfan
10![1008] = Yottagoogolfan
10![1009] = Ronnagoogolfan
10![10010] = Quetagoogolfan
10![100100] = Googolgoogolfan
10![10010![100]] = Googolplexifan
10![(10![100100])100] = Googolplexigoogolfan
These are some extensions and symbols that can be used to make MTSA-FAN more powerful.
a![@b] = a![b, b, b...b, b] with a![b] recursions of b.
a![@b, @c] = a![b, b, b...b, b, c, c, c...c, c] with a![b] entries of b, and a![c] entries of c.
a![@b, @c, @d] = a![b, b, b...b, b, c, c, c...c, c, d, d, d...d, d] with a![b] entries of b, a![c] entries of c, and a![d] entries of d.
and so on, with any number of entries.
a![@@b] = a![@b, @b, @b...@b, @b] with a![@b] entries of @b.
a![@@b, @@c] = a![@b, @b, @b..@b, @b, @c, @c, @c...@c, @c] with a![@b] entries of @b and a![@c] entries of @c.
and so on. This also continues with any number of entries.
a![@@@b] = a![@@b, @@b, @@b...@@b, @@b] with a![@@b] entries of a@@b.
a![@@@b, @@@c] = a![@@b, @@b, @@b...@@b, @@b, @@c, @@c, @@c...@@c, @@c] with a![@@b] entries of a@@b, and a![@@c] entries of @@c.
and so on, with any number of entries, and any number of @'s. The notation can be cleaned up like so:
a![xb] = a![@@...@@b] with x @'s. When notated like a![xbc] the "c" takes effect first, like so: a![@@...@@({{...{{b}}...}})]
10![@10] = 10![10, 10...10, 10] with 10![10] entries = Tenaten
10![@@10] = 10![@10, @10...@10, @10] with Tenaten entries = Tenbitaten
10![@@@10] = 10![@@10, @@10...@@10, @@10] with Tenbitaten entries = Tentritaten
10![10010] = Tenunaten
10![10![10![{{{100}}}]100]100] = Megatenutia
a![&b] = a![bb]
a![&&b] = a![&(bb)] = a![&(@@...@@@b)] = a![b![@@...@@@b]@@...@@@b]
a![&&&b] = a![&&bb] = a![&(b![@@...@@@b]@@...@@@b)] = a![b![b![b![@@...@@@b]@@...@@@b]@@...@@@b]@@...@@@b]
a![&b, &c] = a![bb, cc,]
a![&&b, &&c] = a![&bb, &cc] = a![&(@@...@@@b), &(@@...@@@c)]
and so on. There can be any number of entries with any number of &'s. It can be more easily notated like so:
a![cb] = a![&&...&&b] with c b's.
Basically, when there is an & before an entry, if the entry is just a number, the subscript is that number. If the entry has @(s), then the subscript becomes b![...] where the ... is whatever the original entry is. Example: 3![&@@@7] = 3![7![@@@7]@@@7]
10![&&10] = 10![&1010] = 10![&(@@@@@@@@@@10)]] = 10![10![@@@@@@@@@@10]@@@@@@@@@@10] = Tenbinanden
10![&&&10] = 10![&&1010] = 10![10![10![10![@@@@@@@@@@10]@@@@@@@@@@10]@@@@@@@@@@10]@@@@@@@@@@10] = Tentrinanden
10![1010] = 10![&&&&&&&&&&10] = Tendenanden
10![100010] = Tenkilanden
10![Tenkilanden10] = 10![&&...&&10] with Tenkilanden &'s = Kilotenkilanden. If I am not mistaken, this is my largest googolism on this website.
aѦb = a^b
aѦbѦc = a^^...^^b with c arrows
aѦbѦcѦd = a^^...^^b with cѦd arrows
aѦbѦcѦdѦe = a^^...^^b with cѦdѦe arrows
aѦbѦcѦdѦeѦf = a^^...^^ b with cѦdѦeѦf arrows
and so on.
aѦѦb = aѦaѦaѦa...aѦaѦa with b recursions
aѦѦbѦѦc = aѦaѦa...aѦaѦa with bѦѦc recursions
aѦѦbѦѦcѦѦd = aѦaѦa...aѦaѦa with bѦѦcѦѦd recursions
aѦѦѦb = aѦѦaѦѦa...aѦѦaѦѦa with b recursions
aѦѦѦbѦѦѦc = aѦѦaѦѦa...aѦѦaѦѦa with bѦѦѦc recursions
and so on. You can tell how this expands with more Ѧ's, and more entries.
aѪb = aѦѦ...ѦѦa with b Ѧ's
aѪbѪc = aѦѦ...ѦѦa with bѪc Ѧ's
aѪbѪcѪd = aѦѦ...ѦѦa with bѪcѪd Ѧ's
aѪѪb = aѪaѪa...aѪaѪa with b recursions
aѪѪbѪѪc = aѪaѪa...aѪaѪa with bѪѪc recursions
aѪѪbѪѪcѪѪd = aѪaѪa...aѪaѪa with bѪѪcѪѪd recursions
aѪѪѪb = aѪѪaѪѪa...aѪѪaѪѪa with b recursions
aѪѪѪbѪѪѪc = aѪѪaѪѪa...aѪѪaѪѪa with bѪѪc recursions
and so on and so forth with any number of Ѫ's and any number of entries.
a[Ѫ]b = aѪѪ...ѪѪa with b Ѫ's
a[Ѫ]b[Ѫ]c = aѪѪ..ѪѪa with b[Ѫ]c Ѫ's
a[Ѫ]b[Ѫ]c[Ѫ]d = aѪѪ...ѪѪa with b[Ѫ]c[Ѫ[d] Ѫ's
a[Ѫ][Ѫ]b = a[Ѫ]a[Ѫ]a...a[Ѫ]a[Ѫ] with b recursions
a[Ѫ][Ѫ]b[Ѫ][Ѫ]c = a[Ѫ]a[Ѫ]a...a[Ѫ]a[Ѫ]a with b[Ѫ][Ѫ]c [Ѫ]'s
a[Ѫ][Ѫ][Ѫ]b = a[Ѫ][Ѫ]a[Ѫ][Ѫ]a...a[Ѫ][Ѫ]a[Ѫ][Ѫ]a with b recursions
a[Ѫ][Ѫ][Ѫ]b[Ѫ][Ѫ][Ѫ]c = a[Ѫ][Ѫ]a[Ѫ][Ѫ]a...a[Ѫ][Ѫ]a[Ѫ][Ѫ]a with b[Ѫ][Ѫ]c
with any number of [Ѫ]'s and any number of entries.
a[[Ѫ]]b = a[[]]b = a[Ѫ][Ѫ]...[Ѫ][Ѫ]a with b [Ѫ]
Adding an additional set of [] makes it the next "symbol". There can be any number of []'s, any number of entries, etc., expanding it in the same way.
a{1}b = aѦb
a{2}b = aѪb
a{3}b = a[Ѫ]b
a{4}b = a[[Ѫ]]b
a{c{n}d}b a[[[...[[[Ѫ]]]...]]]b with c{n}d []'s
and so on.
a{{n}}b = a{a{a{...a{a{n}b}b...}b}b}b with a{n}b recursions
a{{{n}}}b = a{{a{{a{{...a{{a{{n}}b}}b...}}b}}b}}b with a{{n}}b recursions
a{n}xb = a{{...{{n}}...}}b with x {}'s
a{n}xb{c}de = a{n}(b{c}de)+x-1b
a{n}xb{c}de{f}gh = a{n}(b{c}de{f}gh)+x-1b{c}(b{c}de{f}gh)+k-1e
with any number of entries, continuing in the same way
a{n}x{1}b = a{n}xa{n}xa{n}x...a{n}xa with b recursions
a{n}x{2}b = a{n}xa{n}xa{n}x...a{n}xa with b{2}b recursions
a{n}x{3}b = a{n}xa{n}xa{n}x...a{n}xa with b{3}b recursions
a{n}x{k}b = a{n}xa{n}xa{n}x...a{n}xa with b{k}b recursions
a{n}x{{1}}b = a{n}xa{n}xa{n}x...a{n}xa with b recursions
a{n}x{{2}}b = a{n}xa{n}xa{n}x...a{n}xa with b{2}{2}b recursions
a{n}x{{3}}b = a{n}xa{n}xa{n}x...a{n}xa with b{3}{3}b recursions
a{n}x{{k}}b = a{n}xa{n}xa{n}x...a{n}xa with b{n}{k}b recursions
a{n}x{{{1}}}b = a{n}xa{n}xa{n}x...a{n}xa with b recursions
a{n}x{{{2}}}b = a{n}xa{n}xa{n}x...a{n}xa with b{{2}}{{2}}b recursions
a{n}x{{{3}}}b = a{n}xa{n}xa{n}x...a{n}xa with b{{3}}{{3}}b recursions
a{n}x{{{k}}}b = a{n}xa{n}xa{n}x...a{n}xa with b{{n}}{{k}}b recursions
a{n}x{k}cb = a{n}xa{n}xa{n}x...a{n}xa with b{n}c-1{k}c-1b
a{n}x{k}c{1}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b recursions
a{n}x{k}c{2}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{2}{2}b recursions
a{n}x{k}c{3}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{3}{3}b recursions
a{n}x{k}c{d}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{d}{d}b recursions
a{n}x{k}c{{1}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b recursions
a{n}x{k}c{{2}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{2}{2}{2}b recursions
a{n}x{k}c{{3}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{3}{3}{3}b recursions
a{n}x{k}c{{d}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{d}{d}{d}b recursions
a{n}x{k}c{{{1}}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b recursions
a{n}x{k}c{{{2}}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{{2}}{{2}}{{2}}b recursions
a{n}x{k}c{{{3}}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{{3}}{{3}}{{3}}b recursions
a{n}x{k}c{{{d}}}b = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{{d}}{{d}}{{d}}b recursions
a{n}x{k}c{d}eb = a{n}x{k}ca{n}x{k}ca{n}x{k}c...a{n}x{k}ca with b{d}e-1{d}e-1{d}e-1b recursions\
continuing on, with any number of {} between a and b.
aꙘb = a{a}a{a}a{a}a...{a}a{a}a{a}aa with b {a}a
aꙘbꙘc =a{a}a{a}a{a}a...{a}a{a}a{a}aa with bꙘc {a}a
aꙘbꙘcꙘd =a{a}a{a}a{a}a...{a}a{a}a{a}aa with bꙘcꙘd {a}a
and so on, with any number of entries.
aꙘꙘb = aꙘaꙘa...aꙘaꙘa with b aꙘ's
aꙘꙘbꙘꙘc = aꙘaꙘa...aꙘaꙘa with bꙘꙘc aꙘ's
aꙘꙘbꙘꙘcꙘꙘd = aꙘaꙘa...aꙘaꙘa with bꙘꙘcꙘꙘd aꙘ's
and so on, with any number of entries
aꙘꙘꙘb = aꙘꙘaꙘꙘa...aꙘꙘaꙘꙘa with b aꙘꙘ's
aꙘꙘꙘbꙘꙘꙘc = aꙘꙘaꙘꙘa...aꙘꙘaꙘꙘa with bꙘꙘꙘc aꙘꙘ's
and so on, with any number of of entries, and any number of Ꙙ's between them
a[Ꙙ]b = aꙘꙘꙘ...ꙘꙘꙘa with b Ꙙ's
a[Ꙙ]b[Ꙙ]c = aꙘꙘꙘ...ꙘꙘꙘa with b[Ꙙ]v Ꙙ's
and so on, with any number of entries
a[Ꙙ][Ꙙ]b = a[Ꙙ]a[Ꙙ]a...a[Ꙙ]a[Ꙙ]a with b a[Ꙙ]'s
a[Ꙙ][Ꙙ]b[Ꙙ][Ꙙ]c = a[Ꙙ]a[Ꙙ]a...a[Ꙙ]a[Ꙙ]a with b[Ꙙ][Ꙙ]c a[Ꙙ]'s
and so on, with any number of entries, and any number of [Ꙙ]'s between them.
a[Ꙙ]xb = a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b with x [Ꙙ]'s
a{[1]}b = a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b with a[Ꙙ]b [Ꙙ]'s
a{[2]}b = a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b with a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b (with a[Ꙙ]b [Ꙙ]'s) [Ꙙ]'s
a{[3]}b = a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b with a{[2]}b [Ꙙ]'s
a{[n]}b = a[Ꙙ][Ꙙ][Ꙙ]...[Ꙙ][Ꙙ][Ꙙ]b with a{[n-1]}b [Ꙙ]'s
a{[[1]]}b = a{[a{[1]}b]}b
a{[[2]]}b = a{[a{[a{[[1]]}b]}b]}b
a{[[3]]}b = a{[a{[a{[a{[[2]]}b]}b]}b]}b
a{[[n]]}b = a{[a{[...a{[a{[[n-1]]}b]}b...]}b]}b with n iterations
a{[[[1]]]}b = a{[[a{[[1]]}]]}b
a{[[[2]]]}b = a{[[a{[[a{[[[1]]]}]]}b]]}b
a{[[[3]]]}b = a{[[a{[[a{[[a{[[[2]]]}b]]}b]]}b]]}b
a{[[[n]]]}b = a{[[a{[[...a{[[a{[[[n-1]]]}b]]}b...]]}b]]}b with n iterations
and so on, with any number of []'s
aꙜb = a{[a]b}b = a{[[[...[[a]]...]]]}b with b []'s
aꙜbꙜc = a{[a]bꙜc}b
aꙜbꙜcꙜd = a{[a]bꙜcꙜd}b
and so on, with any number of entries.
aꙜꙜb = aꙜaꙜa...aꙜaꙜa with b recursions
aꙜꙜbꙜꙜc = aꙜaꙜa...aꙜaꙜa with bꙜꙜc recursions
aꙜꙜbꙜꙜcꙜꙜd = aꙜaꙜa...aꙜaꙜa with bꙜꙜcꙜꙜd recursions
and so on, with any number of entries
aꙜꙜꙜb = aꙜꙜaꙜꙜa...aꙜꙜaꙜꙜa with b recursions
aꙜꙜꙜbꙜꙜꙜc = aꙜꙜaꙜꙜa...aꙜꙜaꙜꙜa with bꙜꙜc recursions
aꙜꙜꙜbꙜꙜꙜcꙜꙜꙜd = aꙜaꙜa...aꙜaꙜa with bꙜꙜcꙜꙜd recursions
and so on, with any number of entries, and any number of Ꙝ's
aꙜxb = aꙜꙜ...ꙜꙜb with x Ꙝ's
I made this to beat my Yus notation, which did not meet my expectations in terms of power. Hopefully this will do better.
⟬a⟭ = a
⟬a, b⟭ = a^b
⟬a, b, c⟭ = a^^...^^b with c arrows
⟬a, b, c, d⟭ = ⟬a, b, ⟬a, b, ⟬a, b, ...⟬a, b, ⟬a, b, c, d-1⟭⟭...⟭⟭⟭ with d recursions
⟬3, 3, 3, 3⟭ = ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, 3, 2⟭⟭⟭⟭ = ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, 3, 1⟭⟭⟭⟭⟭⟭ = ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, ⟬3, 3, 3⟭⟭⟭⟭⟭⟭⟭
⟬a, b, c, d, e⟭ = ⟬a, b, c, ⟬a, b, c, ⟬a, b, c...⟬a, b, c, ⟬a, b, c, d, e-1⟭⟭...⟭⟭⟭ with e recursions
⟬3, 3, 3, 3, 3⟭ = ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, 3, 2⟭⟭⟭⟭ = ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, 3, 1⟭⟭⟭⟭⟭⟭ = ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, ⟬3, 3, 3, 3⟭⟭⟭⟭⟭⟭⟭
⟬a, b, c, d, e, f⟭ = ⟬a, b, c, d, ⟬a, b, c, d, ⟬a, b, c, d...⟬a, b, c, d, ⟬a, b, c, d, e, f-1⟭⟭...⟭⟭⟭ with f recursions
and so on with any number of entries.
⟬a|b⟭ = ⟬a, a, a...a, a, a⟭ with b a's
⟬a|b, c⟭ = ⟬a|⟬a|⟬a|...⟬a|⟬a|b, c-1⟭⟭...⟭⟭⟭ with c recursions
⟬a|b, c, d⟭ = ⟬a|b, ⟬a|b, ⟬a|b...⟬a|b, ⟬a|b, c, d-1⟭⟭...⟭⟭⟭ with d recursions
⟬a|b, c, d, e⟭ = ⟬a|b, c, d, ⟬a|b, c, d, ⟬a|b, c, d...⟬a|b, c, d, ⟬a|b, c, d, e-1⟭⟭...⟭⟭⟭ with e recursions
these arrays can also have any number of entries.
⟬a|b|c⟭ = ⟬a|b, b, b...b, b, b⟭ with c entries of b
⟬a|b|c, d⟭ = ⟬a|b, ⟬a|b, ⟬a|b, ...⟬a|b, ⟬a|b, c, d-1⟭⟭...⟭⟭⟭ with d recursions
⟬a|b|c, d, e⟭ = ⟬a|b, c, ⟬a|b, c, ⟬a|b, c...⟬a|b, c, ⟬a|b, c, d, e-1⟭⟭...⟭⟭⟭ with d recursions
with any number of entries, continuing in the same way
⟬a|b|c|d⟭ = ⟬a|b|c, c, c...c, c, c⟭ with d c's
⟬a|b|c|d, e⟭ = ⟬a|b|c|d, ⟬a|b|c|d, ⟬a|b|c|d...⟬a|b|c|d, ⟬a|b|c|d, e-1⟭⟭...⟭⟭⟭ with e recursions
⟬a|b|c|d, e, f⟭ = ⟬a|b|c|d, e, ⟬a|b|c|d, e, ⟬a|b|c|d, e...⟬a|b|c|d, e, ⟬a|b|c|d, e, f-1⟭⟭...⟭⟭⟭ with f recursions
and so on, with any number of separators and entries
⟬a||b⟭ = ⟬a|a|a...a|a|a⟭ with b separators
⟬a||b, c⟭ = ⟬a||⟬a||⟬a||...⟬a||b, c-1⟭...⟭⟭⟭ with c recursions
⟬a||b, c, d⟭ = ⟬a||b, ⟬a||b, ⟬a||b...⟬a||b, ⟬a||b, c, d-1⟭⟭...⟭⟭⟭ with d recursions
⟬a||b, c, d, e⟭ = ⟬a||b, c, ⟬a||b, c, ⟬a||b, c...⟬a||b,c ⟬a||b, c, d, e-1⟭⟭...⟭⟭⟭ with e recursions
and so on, with any number of entries
⟬a||b||c⟭ = ⟬a||b, b, b...b, b, b⟭ with c entries of b
⟬a||b||c, d⟭ = ⟬a||b, ⟬a||b, ⟬a||b...⟬a||b, ⟬a||b, c, d-1⟭⟭...⟭⟭⟭ with d recursions
⟬a||b||c, d, e⟭ = ⟬a||b, c, ⟬a||b, c, ⟬a||b, c...⟬a||b, c, ⟬a||b, c, d, e-1⟭⟭...⟭⟭⟭ with e recursions
Continuting on, with any number of entries
⟬a||b||c||d⟭ = ⟬a||b||c, c, c...c, c, c⟭ with d entries of c
⟬a||b||c||d, e⟭ = ⟬a|b|c|d, ⟬a|b|c|d, ⟬a|b|c|d...⟬a|b|c|d, ⟬a|b|c|d, e-1⟭⟭...⟭⟭⟭⟭ with e recursions
⟬a||b||c||d, e, f⟭ = ⟬a|b|c|d, e, ⟬a|b|c|d, e, ⟬a|b|c|d, e...⟬a|b|c|d, e, ⟬a|b|c|d, e, f-1⟭⟭...⟭⟭⟭⟭ with f recursions
and so on, with any number of entries
⟬a|||b⟭ = ⟬a||a||a...a||a||a⟭ with b a's
and this continues in the same way, with different amounts of variables, separators, and so on. There can be any number of |'s between entries, and with their own arrays of any number of entries.
⟬a\b⟭ = ⟬a|||...|||a⟭ with b |'s
⟬a\b, c⟭ =⟬a\⟬a\⟬a\...⟬a\⟬a\⟬a\\b, c-1⟭⟭⟭...⟭⟭⟭ with c recursions
⟬a\b, c, d⟭ =⟬a\b, ⟬a\b, ⟬a\b...⟬a\b, ⟬a\b, ⟬a\\b, c, d-1⟭⟭⟭...⟭⟭⟭ with c recursions
and so on, with any number of entries. With any number of \'s, they can have an array after, which works the same way, through recursion and decay.
⟬a\b\c⟭ = ⟬a\b, b, b...b, b⟭ with c entries of b
⟬a\b\c\d⟭ = ⟬a\b\c, c, c...c, c, c⟭ with d entries of c
and so on, with any number of entries
⟬a\\b⟭ = ⟬a\a\a...a\a\a⟭ with b a's
⟬a\\b\\c⟭ = ⟬a\\b, b, b...b, b, b⟭ with c entries of b
⟬a\\b\\c\\d⟭ = ⟬a\\b\\c, c, c...c, c, c⟭ with d entries of c
and so on, with any number of entries
⟬a\\\b⟭ = ⟬a\\a\\a...a\\a\\a⟭ with b a's
⟬a\\\b\\\c⟭ = ⟬a\\\b, b, b...b, b, b⟭ with c b's
⟬a\\\b\\\c\\\d⟭ = ⟬a\\\b\\\c, c, c...c, c, c⟭ with d c's
and so on, with any number of entries, and any number of \s
⟬a&b⟭ = ⟬a\\...\\a⟭ with b \'s
to be expanded
⟦a⟧ = a
⟦a, b⟧ = a^b
⟦a, b, 2⟧ = a^^b = ⟦a, ⟦a, ⟦a...⟦a, ⟦a, a⟧⟧...⟧⟧⟧ b times = ⟦a, ⟦a, ⟦a...⟦a, ⟦a, a, 1⟧, 1⟧, 1...⟧, 1⟧, 1⟧ b times
⟦3, 1, 2⟧ = ⟦3⟧ = 3 = 3^^1
⟦3, 2, 2⟧ = ⟦3, 3⟧ = 3^^2 = 3^3
⟦3, 3, 2⟧ = ⟦3, ⟦3, 3⟧⟧ = 3^3^3 = 3^^3
⟦a, b, 3⟧ = a^^^b =⟦a, ⟦a, ⟦a...⟦a, ⟦a, a, 2⟧, 2⟧, 2...⟧, 2⟧, 2⟧ b times
⟦3, 2, 3⟧ = ⟦3, 3, 2⟧ = ⟦3, ⟦3, 3⟧⟧= 3^3^3 = 3^^^2 = 3^^3
⟦3, 3, 3⟧ = ⟦3, ⟦3, 3, 2⟧, 2⟧ = ⟦3, ⟦3, ⟦3, 3⟧⟧, 2⟧ = ⟦3, 7,625,597,484,987, 2⟧ = 3^^^3 =3^^3^^3
⟦a, b, c⟧ = a^^...^^b with c arrows = ⟦a, ⟦a, ⟦a...⟦a, ⟦a, a, c-1⟧, c-1⟧, c-1...⟧, c-1⟧, c-1⟧ b times
⟦a, b, c, 2⟧ = ⟦a, b, ⟦a, b, ⟦a, b...⟦a, b, ⟦a, b, b⟧⟧...⟧⟧⟧ c times
⟦3, 3, 3, 2⟧ = ⟦3, 3, ⟦3, 3, 3⟧⟧
⟦3, 3, 4, 2⟧ = ⟦3, 3, ⟦3, 3, ⟦3, 3, 3⟧⟧⟧
⟦a, b, c, 3⟧ = ⟦a, b, ⟦a, b, ⟦a, b...⟦a, b, ⟦a, b, b, 2⟧, 2⟧, 2...⟧, 2⟧, 2⟧ c times
⟦3, 3, 3, 3⟧ = ⟦3, 3, ⟦3, 3, 3, 2⟧, 2⟧ = ⟦3, 3, ⟦3, 3, ⟦3, 3, 3⟧⟧, 2⟧
⟦a, b, c, d⟧ = ⟦a, b, ⟦a, b, ⟦a, b...⟦a, b, ⟦a, b, b, d-1⟧, d-1⟧, d-1...⟧, d-1⟧, d-1⟧ c times
⟦a, b, c, d, 2⟧ = ⟦a, b, c, ⟦a, b, c, ⟦a, b, c...⟦a, b, c, ⟦a, b, c, c⟧⟧...⟧⟧⟧ d times
⟦a, b, c, d, 3⟧ = ⟦a, b, c, ⟦a, b, c, ⟦a, b, c...⟦a, b, c, ⟦a, b, c, c, 2⟧, 2⟧...⟧, 2⟧, 2⟧ d times
⟦a, b, c, d, e⟧ = ⟦a, b, c, ⟦a, b, c, ⟦a, b, c...⟦a, b, c, ⟦a, b, c, c, e-1⟧, e-1⟧...⟧, e-1⟧, e-1⟧ d times
⟦a, b, c, d, e, 2⟧ = ⟦a, b, c, d, ⟦a, b, c, d, ⟦a, b, c, d...⟦a, b, c, d, ⟦a, b, c, d, d⟧⟧...⟧⟧⟧ e times
⟦a, b, c, d, e, 3⟧ = ⟦a, b, c, d, ⟦a, b, c, d, ⟦a, b, c, d...⟦a, b, c, d, ⟦a, b, c, d, d, 2⟧, 2⟧...⟧, 2⟧, 2⟧ e times
⟦a, b, c, d, e, f⟧ = ⟦a, b, c, d, ⟦a, b, c, d, ⟦a, b, c, d...⟦a, b, c, d, ⟦a, b, c, d, d, f-1⟧, f-1⟧...⟧, f-1⟧, f-1⟧ e times
and so on, with any number of entries
⟦a/b⟧ = ⟦a, a, a...a, a, a⟧ with b a's
⟦a/b, 2⟧ = ⟦a/⟦a/⟦a/...⟦a/⟦a/a⟧⟧...⟧⟧⟧ b times
⟦a/b, 3⟧ = ⟦a/⟦a/⟦a/...⟦a/⟦a/a, 2⟧, 2⟧... 2⟧, 2⟧, 2⟧ b times
⟦a/b, c⟧ = ⟦a/⟦a/⟦a/⟦...⟦a/⟦a/a, c-1⟧, c-1⟧...⟧, c-1⟧, c-1⟧, c-1⟧
⟦a/b, c, 2⟧ = ⟦a/b, ⟦a/b, ⟦a/b...⟦a/b, ⟦a/b, b⟧⟧...⟧⟧⟧ c times
⟦a/b, c, 3⟧ = ⟦a/b, ⟦a/b, ⟦a/b, ⟦...⟦a/b, ⟦a/b, b, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ c times
⟦a/b, c, d⟧ =⟦a/b, ⟦a/b, ⟦a/b, ⟦...⟦a/b, ⟦a/b, b d-1⟧, d-1⟧...⟧, d-1⟧, d-1⟧, d-1⟧
⟦a/b, c, d, 2⟧ = ⟦a/b, c, ⟦a/b, c, ⟦a/b, c...⟦a/b, c, ⟦a/b, c, c⟧⟧...⟧⟧⟧
⟦a/b, c, d, 3⟧ = ⟦a/b, c, ⟦a/b, c, ⟦a/b, c...⟦a/b, c, ⟦a/b, c, c, 2⟧, 2...⟧, 2⟧, 2⟧
⟦a/b, c, d, e⟧ = ⟦a/b, c, ⟦a/b, c, ⟦a/b, c...⟦a/b, c, ⟦a/b, c, c, e-1⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧
and so on, with any number of entries after the slash.
⟦a/b/2⟧ = ⟦a/a, a, a...a, a, a⟧ with b a's
⟦a/b/3⟧ = ⟦a/⟦a/⟦a/⟦...⟦a/⟦a/a/2⟧/2⟧...⟧/2⟧/2⟧/2⟧ b times
⟦a/b/c⟧ = ⟦a/⟦a/⟦a/⟦...⟦a/⟦a/a/c-1⟧/c-1⟧...⟧/c-1⟧/c-1⟧/c-1⟧ b times
⟦a/b/c, 2⟧ = ⟦a/b/⟦a/b/⟦a/b/...⟦a/b/⟦a/b/b⟧⟧...⟧⟧⟧ c times
⟦a/b/c, 3⟧ = ⟦a/b/⟦a/b/⟦a/b/⟦...⟦a/b/⟦a/b/b, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ c times
⟦a/b/c, d⟧ = ⟦a/b/⟦a/b/⟦a/b/⟦...⟦a/b/⟦a/b/b, d-1⟧, d-1⟧...⟧, d-1⟧, d-1⟧, d-1⟧ c times
⟦a/b/c, d, 2⟧ = ⟦a/b/ c, ⟦a/b/, c, ⟦a/b/c, ⟦...⟦a/b/c, ⟦a/b/c, c⟧⟧...⟧⟧⟧⟧ d times
⟦a/b/c, d, 3⟧ = ⟦a/b/ c, ⟦a/b/, c, ⟦a/b/c, ⟦...⟦a/b/c, ⟦a/b/c, c, 2⟧, 2⟧, 2...⟧, 2⟧, 2⟧, 2⟧d times
⟦a/b/c, d, e⟧ = ⟦a/b/ c, ⟦a/b/, c, ⟦a/b/c, ⟦...⟦a/b/c, ⟦a/b/c, c, e-1⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧, e-1⟧ d times
and so on, with any number of entries of the slash
⟦a/b/c/2⟧ = ⟦a/b/b, b, b...b, b, b⟧ with c b's
⟦a/b/c/3⟧ = ⟦a/b/⟦a/b/⟦a/b/⟦...⟦a/b/⟦a/b/b/2⟧/2⟧...⟧/2⟧/2⟧/2⟧ c times
⟦a/b/c/d⟧ = ⟦a/b/⟦a/b/⟦a/b/⟦...⟦a/b/⟦a/b/b/d-1⟧/d-1⟧...⟧/d-1⟧/d-1⟧/d-1⟧ c times
⟦a/b/c/d, 2⟧ = ⟦a/b/c/⟦a/b/c/⟦a/b/c/⟦...⟦a/b/c, ⟦a/b/c/c⟧⟧...⟧⟧⟧⟧ d times
⟦a/b/c/d, 3⟧ = ⟦a/b/c/⟦a/b/c/⟦a/b/c/⟦...⟦a/b/c, ⟦a/b/c/c, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ d times
⟦a/b/c/d, e⟧ = ⟦a/b/c, ⟦a/b/c, ⟦a/b/c, ⟦...⟦a/b/c, ⟦a/b/c/c⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧, e-1⟧ d times
⟦a/b/c/d, e, 2⟧ = ⟦a/b/c, d, ⟦a/b/c, d, ⟦a/b/c, d, ⟦...⟦a/b/c, d, ⟦a/b/c/c⟧⟧...⟧⟧⟧⟧ e times
⟦a/b/c/d, e, 3⟧ = ⟦a/b/c, d, ⟦a/b/c, d, ⟦a/b/c, d, ⟦...⟦a/b/c, d, ⟦a/b/c/c⟧, 2⟧, 2...⟧, 2⟧, 2⟧, 2⟧ e times
⟦a/b/c/d, e, f⟧ = ⟦a/b/c, d, ⟦a/b/c, d, ⟦a/b/c, d, ⟦...⟦a/b/c, d, ⟦a/b/c/c⟧, f-1⟧, f-1...⟧, f-1⟧, f-1⟧, f-1⟧ e times
and so on, with any number of entries after the slash, and any number of entries with slashes.
⟦a//b⟧ = ⟦a/a/a...a/a/a⟧ with b a's
⟦a//b, 2⟧ = ⟦a//⟦a//⟦a//...⟦a//⟦a//a⟧⟧...⟧⟧⟧ b times
⟦a//b, 3⟧ = ⟦a//⟦a//⟦a//...⟦a//⟦a//a, 2⟧, 2⟧... 2⟧, 2⟧, 2⟧ b times
⟦a//b, c⟧ = ⟦a//⟦a//⟦a//⟦...⟦a//⟦a//a, c-1⟧, c-1⟧...⟧, c-1⟧, c-1⟧, c-1⟧
⟦a//b, c, 2⟧ = ⟦a//b, ⟦a//b, ⟦a//b...⟦a//b, ⟦a//b, b⟧⟧...⟧⟧⟧ c times
⟦a//b, c, 3⟧ = ⟦a//b, ⟦a//b, ⟦a//b, ⟦...⟦a//b, ⟦a//b, b, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ c times
⟦a//b, c, d⟧ =⟦a//b, ⟦a//b, ⟦a//b, ⟦...⟦a//b, ⟦a//b, b d-1⟧, d-1⟧...⟧, d-1⟧, d-1⟧, d-1⟧
⟦a//b, c, d, 2⟧ = ⟦a//b, c, ⟦a//b, c, ⟦a//b, c...⟦a//b, c, ⟦a//b, c, c⟧⟧...⟧⟧⟧
⟦a//b, c, d, 3⟧ = ⟦a//b, c, ⟦a//b, c, ⟦a//b, c...⟦a//b, c, ⟦a//b, c, c, 2⟧, 2...⟧, 2⟧, 2⟧
⟦a//b, c, d, e⟧ = ⟦a//b, c, ⟦a//b, c, ⟦a//b, c...⟦a//b, c, ⟦a//b, c, c, e-1⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧
and so on, with any number of entries after the double slash.
⟦a//b//2⟧ = ⟦a//a, a, a...a, a, a⟧ with b a's
⟦a//b//3⟧ = ⟦a//⟦a//⟦a//⟦...⟦a//⟦a//a//2⟧//2⟧...⟧//2⟧//2⟧//2⟧ b times
⟦a//b//c⟧ = ⟦a//⟦a//⟦a//⟦...⟦a//⟦a//a//c-1⟧//c-1⟧...⟧//c-1⟧//c-1⟧//c-1⟧ b times
⟦a//b//c, 2⟧ = ⟦a//b//⟦a//b//⟦a//b//...⟦a//b//⟦a//b//b⟧⟧...⟧⟧⟧ c times
⟦a//b//c, 3⟧ = ⟦a//b//⟦a//b//⟦a//b//⟦...⟦a//b//⟦a//b//b, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ c times
⟦a//b//c, d⟧ = ⟦a//b//⟦a//b//⟦a//b//⟦...⟦a//b//⟦a//b//b, d-1⟧, d-1⟧...⟧, d-1⟧, d-1⟧, d-1⟧ c times
⟦a//b//c, d, 2⟧ = ⟦a//b// c, ⟦a//b// c, ⟦a//b//c, ⟦...⟦a//b//c, ⟦a//b//c, c⟧⟧...⟧⟧⟧⟧ d times
⟦a//b//c, d, 3⟧ = ⟦a//b// c, ⟦a//b// c, ⟦a//b//c, ⟦...⟦a//b//c, ⟦a//b//c, c, 2⟧, 2⟧, 2...⟧, 2⟧, 2⟧, 2⟧d times
⟦a//b//c, d, e⟧ = ⟦a//b// c, ⟦a//b// c, ⟦a//b//c, ⟦...⟦a//b//c, ⟦a//b//c, c, e-1⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧, e-1⟧ d times
and so on, with any number of entries of the slash
⟦a//b//c//2⟧ = ⟦a//b//b, b, b...b, b, b⟧ with c b's
⟦a//b//c//3⟧ = ⟦a//b//⟦a//b//⟦a//b//⟦...⟦a//b//⟦a//b//b//2⟧//2⟧...⟧//2⟧//2⟧//2⟧ c times
⟦a//b//c//d⟧ = ⟦a//b//⟦a//b//⟦a//b//⟦...⟦a//b//⟦a//b//b//d-1⟧//d-1⟧...⟧//d-1⟧//d-1⟧//d-1⟧ c times
⟦a//b//c//d, 2⟧ = ⟦a//b//c//⟦a//b//c//⟦a//b//c//⟦...⟦a//b//c, ⟦a//b//c//c⟧⟧...⟧⟧⟧⟧ d times
⟦a//b//c//d, 3⟧ = ⟦a//b//c//⟦a//b//c//⟦a//b//c//⟦...⟦a//b//c, ⟦a//b//c//c, 2⟧, 2⟧...⟧, 2⟧, 2⟧, 2⟧ d times
⟦a//b//c//d, e⟧ = ⟦a//b//c, ⟦a//b//c, ⟦a//b//c, ⟦...⟦a//b//c, ⟦a//b//c//c⟧, e-1⟧, e-1...⟧, e-1⟧, e-1⟧, e-1⟧ d times
⟦a//b//c//d, e, 2⟧ = ⟦a//b//c, d, ⟦a//b//c, d, ⟦a//b//c, d, ⟦...⟦a//b//c, d, ⟦a//b//c//c⟧⟧...⟧⟧⟧⟧ e times
⟦a//b//c//d, e, 3⟧ = ⟦a//b//c, d, ⟦a//b//c, d, ⟦a//b//c, d, ⟦...⟦a//b//c, d, ⟦a//b//c//c⟧, 2⟧, 2...⟧, 2⟧, 2⟧, 2⟧ e times
⟦a//b//c//d, e, f⟧ = ⟦a//b//c, d, ⟦a//b//c, d, ⟦a//b//c, d, ⟦...⟦a//b//c, d, ⟦a//b//c//c⟧, f-1⟧, f-1...⟧, f-1⟧, f-1⟧, f-1⟧ e times
and so on, with any number of entries after the slash, and any number of entries with slashes.
⟦a///b⟧ = ⟦a//a//a...a//a//a⟧ with b a's
and so on, with any number of slashes, continuing in the same way.
⟦a&b⟧ = ⟦a//...//a⟧ with b slashes
Thick Arrow Notation, or TAN. This will hopefully be my last notation.
a⤇b = a^b
a⤇b⤇c = a⤇(a⤇(...a⤇(a⤇b⤇c-1)⤇c-1...)⤇c-1)⤇c-1 b times = a^^...^^b with c arrows
a⤇b⤇c⤇d = a⤇b⤇(a⤇b⤇(...a⤇b⤇(a⤇b⤇c⤇d-1)⤇d-1...)⤇d-1)⤇d-1 c times
a⤇b⤇c⤇d⤇e = a⤇b⤇c⤇(a⤇b⤇c⤇(...a⤇b⤇c⤇(a⤇b⤇c⤇d⤇e-1)⤇e-1...)⤇e-1)⤇e-1 d times
and so on, with any number of entries.
a⤇⤇2 = a⤇a⤇a...a⤇a⤇a with a total a's (a-1 total ⤇'s)
a⤇⤇3 = ((...((a⤇⤇2)⤇⤇2)...)⤇⤇2)⤇⤇2 a times
a⤇⤇b = ((...((a⤇⤇b-1)⤇⤇b-1)...)⤇⤇b-1)⤇⤇b-1 a times
a⤇⤇b⤇2 = a⤇⤇(a⤇⤇(...a⤇⤇(a⤇⤇b)...)) with b total a's
a⤇⤇b⤇c =a⤇⤇(a⤇⤇(...a⤇⤇(a⤇⤇b⤇c-1)⤇c-1...)⤇c-1)⤇c-1
a⤇⤇b⤇c⤇2 = a⤇⤇b⤇(a⤇⤇b⤇(...a⤇⤇b⤇(a⤇⤇b⤇c)...)) c times
a⤇⤇b⤇c⤇d = a⤇⤇b⤇(a⤇⤇b⤇(...a⤇⤇b⤇(a⤇⤇b⤇c⤇d-1)⤇d-1...)⤇d-1)⤇d-1
a⤇⤇b⤇c⤇d⤇2 = a⤇⤇b⤇c⤇(a⤇⤇b⤇c⤇(...a⤇⤇b⤇c⤇(a⤇⤇b⤇c⤇d)...)) d times
a⤇⤇b⤇c⤇d⤇e = a⤇⤇b⤇c⤇(a⤇⤇b⤇c⤇(...a⤇⤇b⤇c⤇(a⤇⤇b⤇c⤇d⤇e-1)⤇e-1...)⤇e-1)⤇e-1 d times
and so on, with any number of arrows/entries after the a⤇⤇b
a⤇⤇b⤇⤇2 = a⤇⤇a⤇a⤇a...a⤇a⤇a with b recursions
a⤇⤇b⤇⤇c = a⤇⤇(a⤇⤇(...a⤇⤇(a⤇⤇b⤇⤇c-1)⤇⤇c-1...)⤇⤇c-1)⤇⤇c-1 b times
a⤇⤇b⤇⤇c⤇2 = a⤇⤇b⤇⤇(a⤇⤇b⤇⤇(...a⤇⤇b⤇⤇(a⤇⤇b⤇⤇c)...)) c times
a⤇⤇b⤇⤇c⤇d = a⤇⤇b⤇⤇(a⤇⤇b⤇⤇(...a⤇⤇b⤇⤇(a⤇⤇b⤇⤇c⤇d-1)⤇d-1...)⤇d-1)⤇d-1 c times
a⤇⤇b⤇⤇c⤇d⤇2 = a⤇⤇b⤇⤇c⤇(a⤇⤇b⤇⤇c⤇(...a⤇⤇b⤇⤇c⤇(a⤇⤇b⤇⤇c⤇d)...)) d times
a⤇⤇b⤇⤇c⤇d⤇e = a⤇⤇b⤇⤇c⤇(a⤇⤇b⤇⤇c⤇(...a⤇⤇b⤇⤇c⤇(a⤇⤇b⤇⤇c⤇d⤇e-1)⤇e-1...)⤇e-1)⤇e-1
and so on, with any number of entries/arrows after the a⤇⤇b⤇⤇c
a⤇⤇b⤇⤇c⤇⤇2 = a⤇⤇b⤇⤇c⤇c⤇c...c⤇c⤇c with d c's (and d-1 ⤇'s)
a⤇⤇b⤇⤇c⤇⤇d = a⤇⤇b⤇⤇(a⤇⤇b⤇⤇(...a⤇⤇b⤇⤇(a⤇⤇b⤇⤇c⤇⤇d-1)⤇⤇d-1...)⤇⤇d-1)⤇⤇d-1 d times
a⤇⤇b⤇⤇c⤇⤇d⤇2 = a⤇⤇b⤇⤇c⤇⤇(a⤇⤇b⤇⤇c⤇⤇(...a⤇⤇b⤇⤇c⤇⤇(a⤇⤇b⤇⤇c⤇⤇d)...)) d times
a⤇⤇b⤇⤇c⤇⤇d⤇e = a⤇⤇b⤇⤇c⤇⤇(a⤇⤇b⤇⤇c⤇⤇(...a⤇⤇b⤇⤇c⤇⤇(a⤇⤇b⤇⤇c⤇⤇d⤇e-1)⤇e-1...)⤇e-1)⤇e-1 d times
and so on, win any number of entries with single arrows and double arrows.
a⤇⤇⤇b = a⤇⤇a⤇⤇a...a⤇⤇a⤇⤇a with b a's
triple arrow works in the same way as double arrows, except with another arrow.
a⤇⤇⤇b⤇⤇c⤇d, for example, = a⤇⤇⤇b⤇⤇(a⤇⤇⤇b....⤇d-1)⤇d-1
a⤇⤇⤇⤇b = a⤇⤇⤇a⤇⤇⤇a...a⤇⤇⤇a⤇⤇⤇a b times
and so on, continuing in the same way. There can be any number of arrows.
a⤇[2] = a⤇⤇⤇...⤇⤇⤇a with a arrows
a⤇[3] = ((...(a⤇[2])⤇[2])⤇[2]...)⤇[2])⤇[2] a times
a⤇[b] = ((...(a⤇[b-1])⤇[b-1])[b-1]...)⤇[b-1])⤇[b-1] a times
a⤇[b, 2] = a⤇[a⤇[...a⤇[b]...]] with b a's
a⤇[b, c] = a⤇[a⤇[...a⤇[b, c-1], c-1...], c-1] with b a's
a⤇[b, c, 2] = a⤇[b, a⤇[b, ...a⤇[b, b]...]] with c a's
a⤇[b, c, d] = a⤇[b, a⤇[b, a⤇[....a⤇[b, a, d-1], d-1....], d-1], d-1] with c a's
and so on, with any number of entries in the array.
a⤇[b]⤇[2] = a⤇[a, a, a...a, a, a] with b a's in the array
a⤇[b]⤇[c] = a⤇[a⤇[...a⤇[a⤇[b]⤇[c-1]]⤇[c-1]...]⤇[c-1]]⤇[c-1] with b a's
a⤇[b]⤇[c, 2] = a⤇[b]⤇[a⤇[b]⤇[...a⤇[b]⤇[a⤇[b]]...]] with c a's
a⤇[b]⤇[c, d] = a⤇[b]⤇[a⤇[b]⤇[...a⤇[b]⤇[a⤇[b, d-1], d-1], d-1...], d-1] with c a's
a⤇[b]⤇[c, d, e] = a⤇[b]⤇[c, a⤇[b]⤇[c, ...a⤇[b]⤇[c, a⤇[c, c, e-1], e-1], e-1...], e-1] with d a's
and so on, with any number of entries in the array, and any number of a⤇[b]⤇[c]⤇[d]⤇[e]..........
a⤇⤇[2] = a⤇[a]⤇[a]⤇[a]...[a]⤇[a]⤇[a] with a [a]'s
a⤇⤇[3] = ((...(a⤇⤇[2])⤇⤇[2])⤇⤇[2]...)⤇⤇[2])⤇⤇[2] a times
a⤇⤇[b] = ((...(a⤇⤇[b-1])⤇⤇[b-1])⤇⤇[b-1]...)⤇⤇[b-1])⤇⤇[b-1] a times
a⤇⤇[b, .....] works in the same way as a⤇[b, .....] except with two arrows instead of one.
and a⤇⤇⤇[b] works in the same way except with three arrows.
a⤇[[2]] = a⤇⤇⤇...⤇⤇⤇[a] with a arrows
a⤇[[b]] = ((...(a⤇[[b-1]])⤇[[b-1]])[[b-1]]...)⤇[[b-1]])⤇[[b-1]] a times
a⤇⤇...⤇⤇[[b]]⤇.... works the same way as it does with just one pair of brackets, except with two.
a⤇⤇...⤇⤇[[[b]]]⤇.... again, works in the same way but with an extra pair of brackets
There can be any number of brackets around the variable.
a⤇{2} = a⤇[[...[[a]]...]] with b pairs of brackets.
a⤇{3} = ((...((a⤇{2})⤇{2})⤇{2}...⤇{2})⤇{2})⤇{2}
a⤇{b} = ((...((a⤇{b-1})⤇{b-1})⤇{b-1}...⤇{b-1})⤇{b-1})⤇{b-1} with a a's
a⤇{b, 2} = a⤇{a⤇{a⤇{...a⤇{a⤇{b}}...}}} with b a's
a⤇{b, 3} = a⤇{a⤇{a⤇{...a⤇{a⤇{b}, 2}, 2...}, 2}, 2} with b a's
a⤇{b, c} = a⤇{a⤇{a⤇{...a⤇{b}, c-1...}, c-1}, c-1} with b a's
a⤇{b, c, 2} = a⤇{b, a⤇{b, ...a⤇{b, a⤇{b, c}}...}} with c a's
a⤇{b, c, d} = a⤇{b, a⤇{b, a⤇{...a⤇{b, a⤇{b, c, d-1}, d-1}...}, d-1}, d-1} with c a's
a⤇{b, c, d, 2} = a⤇{b, c, a⤇{b, c, a⤇{...b, c, a⤇{b, c, d}...}}} with d a's
a⤇{b, c, d, e} = a⤇{b, c, a⤇{b, c, a⤇{...b, c, a⤇{b, c, d, e-1}, e-1...}, e-1}, e-1} with d a's
and so on, with any number of entries.
This repeats in the same way as it does with [], except with {}. a⤇[b] = a⤇[b]1, a⤇[b]2 = a⤇{b}, and a⤇[b]3 is naturally the next step, and so on.
this may be expanded in the future
☾a☽ = a
☾a, b☽ = ☾a, b, 1☽ = a^b
☾a, b, 2☽ = ☾a, ☾a, ☾a...☾a, a☽...☽☽☽ with ☾b, b☽ a's
☾a, b, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 2☽, 2☽...☽, 2☽, 2☽ with ☾b, b, 2☽ a's
☾a, b, c☽ = ☾a, ☾a, ☾...☾a, ☾a, a c-1☽, c-1☽...☽, c-1☽, c-1☽ with ☾b, b, c-1☽ a's/recursions
☾a, b, 1, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, a☽☽...☽☽☽ with ☾b, b, b☽ recursions
☾a, b, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2☽, 1, 2☽, 1, 2...☽, 1, 2☽, 1, 2☽ with ☾b, b, 1, 2☽ recursions
☾a, b, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 2, 2☽, 2, 2☽, 2, 2...☽, 2, 2☽, 2, 2☽ with ☾b, b, 2, 2☽ recursions
☾a, b, c, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 2☽, c-1, 2☽, c-1, 2...☽, c-1, 2☽, c-1, 2☽ with ☾b, b, c-1, 2☽ recursions
☾a, b, 1, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2☽, 1, 2☽, 1, 2...☽, 1, 2☽, 1, 2☽ with ☾b, b, 1, 2☽ recursions
☾a, b, 2, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 3☽, 1, 3☽, 1, 3...☽, 1, 3☽, 1, 3☽ with ☾b, b, 1, 3☽ recursions
☾a, b, c, 3☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 3☽, c-1, 3☽, c-1, 3...☽, c-1, 3☽, c-1, 3☽ with ☾b, b, c-1, 3☽ recursions
☾a, b, c, d☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d☽, c-1, d☽, c-1, d...☽, c-1, d☽, c-1, d☽ with ☾b, b, c-1, d☽ recursions
☾a, b, 1, 1, 2☽ = ☾a, a, a, ☾a, a, a, ☾...☾a, a, a, ☾a, a, a, a☽☽...☽☽☽ with ☾b, b, b, b☽ recursions
☾a, b, 2, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 1, 2☽, 1, 1, 2☽...☽, 1, 1, 2☽, 1, 1, 2☽ with ☾b, b, 1, 1, 2☽ recursions
☾a, b, c, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 1, 2☽ c-1, 1, 2☽...☽, c-1, 1, 2☽, c-1, 1, 2☽ with ☾b, b, c-1, 1, 2☽ recursions
☾a, b, 1, 2, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, 1, 1, 2☽, 1, 2☽...☽ 1, 2☽, 1, 2☽ with ☾b, b, 1, 1, 2☽ recursions
☾a, b, 2, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 2, 2☽, 1, 2, 2☽...☽, 1, 2, 2☽, 1, 2, 2☽ with ☾b, b, 1, 2, 2☽ recursions
☾a, b, c, 2, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 2, 2☽, c-1, 2, 2☽...☽, c-1, 2, 2☽, c-1, 2, 2☽ with ☾b, b, c-1, 2, 2☽ recursions
☾a, b, 1, 3, 2☽ = ☾a, a, ☾a, a, ☾...☾a, a, ☾a, a, 1, 2, 2☽, 2, 2☽...☽, 2, 2☽, 2, 2☽ with ☾b, b, 1, 2, 2☽ recursions
☾a, b, 2, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 3, 2☽, 1, 3, 2☽...☽, 1, 3, 2☽ 1, 3, 2☽ with ☾b, b, 1, 2, 2☽ recursions
☾a, b, c, 3, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, 3, 2☽, c-1, 3, 2☽...☽, c-1, 3, 2☽, c-1, 3, 2☽ with ☾b, b, c-1, 3, 2☽ recursions
☾a, b, c, d, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, 2☽, c-1, d, 2☽...☽ c-1, d, 2☽, c-1, d, 2☽ with ☾b, b, c-1, d, 2☽ recursions
☾a, b, c, d, e☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, e☽, c-1, d, e☽...☽ c-1, d, e☽, c-1, d, e☽ with ☾b, b, c-1, d, e☽ recursions
☾a, b, 1, 1, 1, 2☽ = ☾a, a, a, a, ☾a, a, a, a, ☾...☾a, a, a, a, ☾a, a, a, a, a☽☽...☽☽☽ with ☾b, b, b, b, b☽ recursions
☾a, b, 2, 1, 1, 2☽ = ☾a, ☾a, ☾...☾a, ☾a, a, 1, 1, 1, 2☽, 1, 1, 1, 2☽...☽ 1, 1, 1, 2☽, 1, 1, 1, 2☽ with ☾b, b, 1, 1, 1, 2☽ recursions
☾a, b, c, d, e, f☽ = ☾a, ☾a, ☾...☾a, ☾a, a, c-1, d, e, f☽, c-1, d, e, f☽...☽ c-1, d, e, f☽, c-1, d, e, f☽ with ☾b, b, c-1, d, e, f☽ recursions
and so on, with any number of entries.
☾a|2☽ = ☾a, a, a...a, a, a☽ with a entries of a
☾a|3☽ = ☾☾☾...☾a|2☽|2...☽|2☽|2☽ with ☾a|2☽ recursions
☾a|b☽ = ☾☾☾...☾a|b-1☽|b-1...☽|b-1☽|b-1☽ with ☾a|b-1☽ recursions
☾a|1, 2☽ = ☾☾☾...☾☾a|a☽|a☽...☽|a☽|a☽ with ☾a|a☽ a's
☾a|2, 2☽ = ☾☾☾...☾☾a|1, 2☽|1, 2☽...☽|1, 2☽|1, 2☽ with ☾a|1,2☽ ☾...|1,2☽'s
☾a|b, 2☽ = ☾☾☾...☾☾a|b-1, 2☽|b-1, 2☽...☽|b-1, 2☽|b-1, 2☽ with ☾a|b-1,2☽ ☾...|b-1,2☽'s
☾a|b, c☽ = ☾☾☾...☾☾a|b-1, c☽|b-1, c☽...☽|b-1, c☽|b-1, c☽ with ☾a|b-1, c☽ ☾...|\b-1,c☽'s
☾a|1, 1, 2☽ = ☾☾☾...☾☾a|a, a☽|a, a☽...☽|a, a☽|a, a☽ with ☾a|a, a☽ recursions
☾a|b, c, d☽ = ☾☾☾...☾☾a|b-1, c, d☽|b-1, c, d☽...☽|b-1, c, d☽|b-1, c, d☽ with ☾a|b-1, c, d☽ recursions
and so on, with any number of entries.
☾a||2☽ = ☾a|a, a, a...a, a☽ with a entries of a
☾a||b☽ = ☾☾☾....☾☾a||b-1☽||b-1☽...☽||b-1☽||b-1☽ with ☾a||b-1☽ recursions
☾a||1, 2☽ = ☾☾☾...☾☾a||a☽||a☽...☽||a☽||a☽ with ☾a||a☽ recursions
☾a||b, 2☽ = ☾☾☾....☾☾a||b-1, 2☽||b-1, 2☽...☽||b-1, 2☽||b-1, 2☽ with ☾a||b-1, 2☽ recursions
☾a||b, c☽ = ☾☾☾...☾☾a||b-1, c☽||b-1, c☽...☽||b-1, c☽||b-1, c☽ with ☾a||b-1, c☽ recursions
☾a||1, 1, 2☽ = ☾☾☾...☾☾a||a, a☽||a, a☽...☽||a, a☽||a, a☽ with ☾a||a, a☽ recursions
☾a||b, c, d☽ = ☾☾☾...☾☾a||b-1, c, d☽||b-1, c, d☽...☽||b-1, c, d☽||b-1, c, d☽ with ☾a||b-1, c, d☽ recursions
and so on, with any number of entries.
☾a|||b...☽ works in the same way as ☾a||b...☽ but with three |'s instead of two. And so on, with any number of separators.
☾a|1|2☽ = ☾a||...||a☽ with a separators = ☾a/2☽
☾a|2|2☽ = ☾☾☾...☾☾a|1|2☽|1|2☽...☽|1|2☽|1|2☽ with ☾a|1|2☽ recursions
☾a|b|2☽ = ☾☾☾...☾☾a|b-1|2☽|b-1|2☽...☽|b-1|2☽|b-1|2☽ with ☾a|b-1|2☽ recursions = ☾a/b☽
☾a|b|c☽ = ☾☾☾...☾☾a|b-1|c☽|b-1|c☽...☽|b-1|c☽|b-1|c☽ with ☾a|b-1|c☽ recursions
☾a|1, 1|2☽ ☾a|a|☾a|a|☾...☾a|a|☾a|a|a☽☽...☽☽☽ with ☾a|a|a☽ recursions
☾a|2, 1|2☽ = ☾☾☾...☾☾a|1, 1|2☽|1, 1|2☽...☽|1, 1|2☽|1, 1|2☽ with ☾a|1, 1|2☽ recursions
☾a|b, 1|2☽ = ☾☾☾...☾☾a|b-1, 1|2☽|b-1, 1, 2☽...☽|b-1, 1, 2☽|b-1, 1|2☽
☾a|1, 2|2☽ = ☾☾☾...☾☾a|a, 1|2☽|a, 1, 2☽...☽|a, 1|2☽|a, 1|2☽ with ☾a|a, 1|2☽ recursions
☾a|2, 2|2☽ = ☾☾☾...☾☾a|1, 2|2☽|1, 2|2☽...☽|1, 2|2☽|1, 2|2☽ with ☾a|1, 2|2☽ recursions
☾a|b, 2|2☽ = ☾☾☾...☾☾a|b-1,2|2☽|b-1, 2|2☽...☽|b-1, 2|2☽|b-1, 2|2☽ with ☾a|b-1,2|2☽ recursions
☾a|b, c|2☽ = ☾☾☾...☾☾a|b-1,c|2☽|b-1, c|2☽...☽|b-1, c|2☽|b-1, c|2☽ with ☾a|b-1, c|2☽ recursions
☾a|b, c|d☽ = ☾☾☾...☾☾a|b-1, c|d☽|b-1, c|d☽...☽|b-1, c|d☽|b-1, c|d☽ with ☾a|b-1, c|d☽ recursions
☾a|b, c, d|e☽ = ☾☾☾...☾☾a|b-1, c, d|e☽|b-1, c, d|e☽...☽|b-1, c, d|e☽|b-1, c, d|e☽ with ☾a|b-1, c, d|e☽ recursions
and so on, with any number of entries after the seperator
☾a|1|1, 2☽ = ☾a|a, a, a...a, a, a|a☽ with a entries of a in the middle
☾a|2|1, 2☽ = ☾☾☾...☾☾a|1|1, 2☽|1|1, 2☽...☽|1|1, 2☽|1|1, 2☽ with ☾a|1|1, 2☽ recursions
☾a|b|1, 2☽ = ☾☾☾...☾☾a|b-1|1, 2☽|b-1|1, 2☽...☽|b-1|1, 2☽|b-1|1, 2☽ with ☾a|b-1|1, 2☽ recursions
☾a|1|2, 2☽ = ☾☾☾...☾☾a|a|1, 2☽|a|1, 2☽...☽|a|1, 2☽|a|1, 2☽ with ☾a|a|1, 2☽ recursions
☾a|2|2, 2☽ = ☾☾☾...☾☾a|1|2, 2☽|1|2, 2☽...☽|1|2, 2☽|1|2, 2☽ with ☾a|1|2, 2☽ recursions
☾a|b|2, 2☽ = ☾☾☾...☾☾a|b-1|2, 2☽|b-1|2, 2☽...☽|b-1|2, 2☽|b-1|2, 2☽ with ☾a|b-1|2, 2☽ recursions
☾a|b|c, d☽ = ☾☾☾....☾☾a|b-1|c, d☽|b-1|c, d☽...☽|b-1|c, d☽|b-1|c, d☽ with ☾a|b-1|c, d☽ recursions
☾a|1|1, 1, 2☽ = ☾☾☾...☾☾a|a|a, a☽|a|a, a☽...☽|a|a, a☽|a|a, a☽ with ☾a|a|a, a☽ recursions
☾a|2|1, 1, 2☽ = ☾☾☾...☾☾a|1|1, 1, 2☽|1|1, 1, 2☽...☽|1|1, 1, 2☽|1|1, 1, 2☽ with ☾a|1|1, 1, 2☽ recursions
☾a|b|c, d, e☽ = ☾☾☾...☾☾a|b-1|c, d, e☽|b-1|c, d, e☽...☽|b-1|c, d, e☽|b-1|c, d, e☽ with ☾a|b-1|c, d, e☽ recursions
and so on, win any number of entries after the separator.
☾a|b...|c...☽ works with any number of entries in each. ☾a|b...|c...|d...|e...☽ any number of separators with any number of entries.
It also works with ☾a||b...||c...||d...☽ which works the same as with one separator between each, except with two.
☾a||1, 1||2☽ = ☾a|a, a, a...a, a, a|a, a, a...a, a, a|a, a, a......a, a, a☽ with a separators, and a a's between each.
this continues on with ☾a|||b...|||c....|||d...☽ and with 4, 5, and any number of separators with any number of entries between them.
☾a:2☽ = ☾a||...||a, a...a, a||...||a, a...a, a||...||......a, a||...||a, a...a, a☽ with a groups of a separators, with a a's between each group.
☾a:b☽ = ☾☾☾...☾☾a:b-1☽:b-1☽...☽:b-1☽:b-1☽ with ☾a:b-1☽ recursions.
☾a:b:2☽ = ☾a:☾a:☾...☾a:☾a:a☽☽...☽☽☽ with ☾b:b☽ recursions
☾a:b:c☽ = ☾a:☾a:☾...☾a:☾a:a:c-1☽:c-1☽...☽:c-1☽:c-1☽ with ☾b:b:c-1☽ recursions
☾a:b:1:2☽ = ☾a:a:☾a:a:☾...☾a:a:☾a:a:a☽☽...☽☽☽ with ☾b:b:b☽ recursions
☾a:b:2:2☽ = ☾a:☾a:☾...☾a:☾a:a:1:2☽:1:2☽...☽:1:2☽:1:2☽ with ☾a:a:1:2☽ recursions
☾a:b:c:2☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:2☽:c-1:2☽...☽:c-1:2☽:c-1:2☽ with ☾a:a:c-1:2☽ recursions
☾a:b:1:3☽ = ☾a:a:☾a:a:☾...☾a:a:☾a:a:a:2☽:2☽...☽:2☽:2☽ with ☾a:a:a:2☽ recursions
☾a:b:c:d☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:d☽:c-1:d☽...☽:c-1:d☽:c-1:d☽ with ☾a:a:c-1:d☽ recursions
☾a:b:1:1:2☽ = ☾a:a:a:☾a:a:a:☾...☾a:a:a:☾a:a:a:a☽☽...☽☽☽ with ☾a:a:a:a☽ recursions
☾a:b:c:d:e☽ = ☾a:☾a:☾...☾a:☾a:a:c-1:d:e☽:c-1:d:e☽...☽:c-1:d:e☽:c-1:d:e☽ with ☾a:a:c-1:d:e☽ recursions
and so on, with any number of entries.
☾a::2☽ = ☾a:a:a...a:a:a☽ with a a's
☾a::b☽ = ☾☾☾...☾☾a::b-1☽:b-1☽...☽:b-1☽:b-1☽ with ☾a::b-1☽ recursions
This continues in the same way as ☾a:b...☽ but with two :'s instead of one.
It continues in the same way with ☾a:::b☽, ☾a::::b☽, and so on.
☾a%2☽ = ☾a:::...:::a☽ with a :'s
☾a%b☽ = ☾☾☾...☾☾a%b-1☽%b-1☽...☽%b-1☽%b-1☽ with ☾a%b-1☽ recursions
☾a%b...☽ continues the same way as ☾a:b☽, except with % instead of :
☾a?☽ = ☾a%a-1%a-2...%4%3%2☽
☾a?2☽ = ☾☾☾...☾☾a?☽?☽...☽?☽?☽ with a ?'s
☾a?b☽ = ☾☾☾...☾☾a?b-1☽?b-1☽...☽?b-1☽?b-1☽ with ☾a?b-1☽ recursions
~(a)~ = 10^a
~(a,2)~ = ~(~(~(...~(~(a)~)~...)~)~)~ with a sets of ~()~
~(3,2)~ = ~(~(~(3)~)~)~ = 10^(10^(10^3)) = 10^(10^1000)
~(a,b)~ = ~(~(~(~(~(~(...~(~(~(~(a,b-1)~,b-1)~)~,b-1)~...)~,b-1)~)~,b-1)~)~,b-1)~ with a b-1's
~(3,3)~ = ~(~(~(3,2)~,2)~,2)~ = ~(~(10^(10^1000), 2)~,2)~
~(5,4)~ = ~(~(~(~(~(5,3)~,3)~,3)~,3)~,3)~
~(a,1,2)~ = ~(a,~(a,~(a,...~(a,~(a,~(a,a)~)~)~...)~)~)~ with a nestings
~(a,b,2)~ = ~(~(~(...~(~(a,b-1,2)~,b-1,2)~...,b-1,2)~,b-1,2)~,b-1,2)~ with a nestings
~(a,1,c)~ = ~(a,~(a,~(a,~(...~(a,~(a,a,c-1)~,c-1)~...)~,c-1)~,c-1)~,c-1)~ with a nestings
~(a,b,c)~ = ~(~(~(~(...~(~(a,b-1,c)~,b-1,c)~...)~,b-1,c)~,b-1,c)~,b-1,c)~ with a nestings
~(a,1,1,2)~ = ~(a,a,~(a,a,~(a,a,...~(a,a,~(a,a,a)~)~...)~)~)~ with a nestings
~(a,b,1,2)~ = ~(~(~(~(...~(~(a,b-1,1,2)~,b-1,1,2)~...)~,b-1,1,2)~,b-1,1,2)~,b-1,1,2)~ with a nestings
~(a,1,c,2)~ = ~(a,~(a,~(a,~(...~(a,~(a,a,c-1,2)~,c-1,2)~...)~c-1,2)~,c-1,2)~,c-1,2)~ with a nestings
~(a,1,1,d)~ = ~(a,1,~(a,1,~(a,1,~(...~(a,1,~(a,1,a,d-1)~,d-1)~...)~,d-1)~,d-1)~,d-1)~ with a nestings
~(a,b,c,d)~ = ~(~(~(~(...~(~(a,b-1,c,d)~,b-1,c,d)~...)~,b-1,c,d)~,b-1,c,d)~,b-1,c,d)~ with a nestings
continuing in the same way for any number of entries.
~(a#2)~ = ~(a, a, a...a, a, a)~ with a a's
~(a#b)~ = ~(~(~(~(...~(~(a#b-1)~#b-1)~...)~#b-1)~#b-1)~#b-1)~ with a nestings
Also called CLAN
First and foremost: lowercase letters are variables, uppercase letters are functions.
Letters are grouped into clusters of 5. (Symbols are to the end)
(A-E), (F-J), (K-O), (P-T), (U-Y), (Z?@#$), (%&*~☆), and (¢£€₰⟰)
Each letter in a cluster shares definition, but shifted. The definition is based off the last digit of n (the last digit is referred to as x) (decimals are ignored)
For example: For A:
if x=0, n+1
if x=1, n&n (using BEAF notation)
if x = 2, n^2
if x = 3, [3]GOSS(n) (using the GOSS function)
if x = 4, n!
if x = 5, <[n]> (using Brick notation)
if x = 6, 2n
if x = 7, HG([n||n]) (using Hexagraphs)
if x = 8, PDN(n) (using Power Digit Number function)
if x = 9, fΓ0(n) (fast-growing hierarchy)
for b, this shifts over 1:
if x=1, n+1
if x=2, n&n (using BEAF notation)
if x = 3, n^2
if x = 4, [3]GOSS(n) (using the GOSS function)
if x = 5, n+3 and add one of each letter to the end of the string
if x = 6, <[n]> (using Brick notation)
if x = 7, 2n
if x = 8, HG([n||n]) (using Hexagraphs)
if x = 9, PDN(n) (using Power Digit Number function)
if x = 0, fΓ0(n) (fast-growing hierarchy)
this shifts over by 1 for each letter in the cluster.
For F:
if x = 0, ~(n#n)~ (using Grilliard Array Notation)
if x = 1, n#n! (using Macro-Factorial notation)
if x = 2, n+6 and add the whole string of letters to the end of the current string
if x = 3, PN(n) (using Pyraginal numbers)
if x = 4, nFx(n) (using FixFactorials)
if x = 5, n^^n
if x = 6, n{n}n
if x = 7, n*4 and switch all "A"s and "M"s in the sequence
if x = 8, ((...((n!)!)!...)!)! with n factorials
if x = 9, GOSS(n)
and again, this is shifted for each letter in the cluster.
For K:
if x = 0, n&n&n (using BEAF notation)
if x = 1, n/2
if x = 2, n![2] (using Hyper-Factorial Array notation)
if x = 3, n-1
if x = 4, fφ(1,0,0,0)(n)
if x = 5, n/(n-2)
if x = 6, n!
if x = 7, = n*3
if x = 8, n+8
if x = 9, n{7}n
For P:
x = 0, (2n)+11 and add one of each letter in order to the end of the string
x = 1, (3n)+12 and add n %s to the end of the string
x = 2, (4n)+13 and add n &s to the end of the string
x = 3, (5n)+14 and add n{n}n of each letter to the end of the string (nAA...AABB...BBCC...CCDD.............⟰⟰)
x = 4, n^100
x = 5, n/(n/2)
x = 6, add n ⟰s to the end of the string
x = 7, add n⟰⟰⟰⟰⟰ Ks to the end of the string
x = 8, n☆☆☆
x = 9, nABCDEFGHIJKLMNOPQRSTUVWXYZ
For U:
x = 0, nP
x = 1, n-1
x = 2, n{3}n, and add 11 of the n%40th letter to the end of the string (n%40 = where % means remainder, not the letter)
x = 3, n{n{n{n}n}n}n, and add n{n}n of the n%40th letter to the end of the string
x = 4, TREE[n] and add the n%40th letter to the end of the string
x = 5, BB(n) and add the n%40th letter to the end of the string
x = 6, Rayo(n) and add the n%40th letter to the end of the string
x = 7, nV and add the n%40th letter to the end of the string
x = 8, n-(n/3)
x = 9, <[n]>/<[n]>/<[n]> in brick notation and add <[n]> of the n%40th letter to the end of the string
For Z:
x = 0, nC
x = 1, n-12
x = 2, n{nK}n
x = 3, n{n{n{n}n}n}n
x = 4, Graham(n) and add the n%40th letter to the end of the string
x = 5, BB(BB(BB(n)))
x = 6, n*6, and swicth all Ts and Ms in the string
x = 7, nN and add the current string to the end of the current string
x = 8, n!
x = 9, Course([n]), and add n Ws to the end of the string
For %:
x = 0, ~(n, 3)~
x = 1, [n]GOSS(n)
x = 2, GODS(n)
x = 3, n in a n-gon using Steinhaus-Moser
x = 4, Goodstein(n)
x = 5, Goodstein(Goodstein(Goodstein(n)))
x = 6, nH
x = 7, Rayo(Rayo(Rayo(n)))
x = 8, n^(nL)
x = 9, {n, 1, 1, 1, 1, 3, 4, 1, 1, 1, 6, 1, 7} and add the n%40th letter to the end of the string
For ¢:
x = 0, ~(n, n, 2)~
x = 1, nF and double up each letter in the sequence (for example nAFFH becomes nAAFFFFHH)
x = 2, HMG(n)
x = 3, HG(n)
x = 4, Quorvask(n)
x = 5, n![n, n, n, n, n, n, n, n, n, n, n]
x = 6, (n!)!
x = 7, n!3
x = 8, ((n!)!)!
x = 9, {n, n, n, n, n, n, n, n, n, n, n, n, n, n, n} and add the n%40th letter to the end of the string
once again, this is shifted by 1 for each letter in any cluster.
Letters work like functions, so for example think of "331ABHFG" as (((((331)A)B)H)F)G
[0]↤[a] = a+1
[0]↤[3] = 4
[1]↤[a] = [0]↤[[0]↤[...[0]↤[[0]↤[a]]...]] with a nestings = 2a
[1]↤[3] = [0]↤[[0]↤[[0]↤[3]]] = [0]↤[[0]↤[4]] = [0]↤[5] = 6
[2]↤[a] = [1]↤[[1]↤[...[1]↤[[1]↤[a]]...]] with a nestings = a*(2^a)
[2]↤[2] = 8
[2]↤[3] = [1]↤[[1]↤[[1]↤[3]]] = [1]↤[[1]↤[6]] = [1]↤[12] = 24
[3]↤[a] = [2]↤[[2]↤[...[2]↤[[2]↤[a]]...]] with a nestings
[3]↤[3] = [2]↤[[2]↤[[2]↤[3]]] = [2]↤[[2]↤[24]] = [2]↤[[2]↤[24]] = [2]↤[402,653,184] = Too big
[3]↤[2] = [2]↤[[2]↤[2]] = [2]↤[8] = 2,048
[4]↤[a] = [3]↤[[3]↤[...[3]↤[[3]↤[a]]...]] with a nestings
[4]↤[3] = [3]↤[[3]↤[[3]↤[3]]]
[4]↤[2] = [3]↤[[3]↤[2]] = [3]↤[2,048]
[b]↤[a] = [b-1]↤[[b-1]↤[...[b-1]↤[[b-1]↤[a]]...]] with a nestings
[5]↤[3] = [4]↤[[4]↤[[4]↤[3]]] = 4↤[[4]↤[3]↤[[3]↤[[3]↤[3]]]] = 4↤[[4]↤[3]↤[[3]↤[[2]↤[402,653,184]]]]
[0]↤[0]↤[a] = [[...[[a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[0]↤[0]↤[3] = [[[3]↤[3]]↤[3]]↤[3] = [[[2]↤[402,653,184]]↤[3]]↤[3]
[0]↤[1]↤[a] = [0]↤[0]↤[[0]↤[0]↤[...[0]↤[0]↤[[0]↤[0]↤[a]]...]] with a nestings
[0]↤[1]↤[3] = [0]↤[0]↤[[0]↤[0]↤[[0]↤[0]↤[3]]]] = [0]↤[0]↤[[0]↤[0]↤[[[[2]↤[402,653,184]]↤[3]]↤[3]]]]
[0]↤[b]↤[a] = [0]↤[b-1]↤[[0]↤[b-1]↤[...[0]↤[b-1]↤[[0]↤[b-1]↤[a]]...]] with a nestings
[0]↤[2]↤[3] = [0]↤[1]↤[[0]↤[1]↤[[0]↤[1]↤[3]]] = [0]↤[1]↤[[0]↤[1]↤[[0]↤[0]↤[[0]↤[0]↤[[[[2]↤[402,653,184]]↤[3]]↤[3]]]]]]
[0]↤[3]↤[3] = [0]↤[2]↤[[0]↤[2]↤[[0]↤[2]↤[3]]]
[1]↤[0]↤[a] = [0]↤[[0]↤[...[0]↤[[0]↤[a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[1]↤[0]↤[3] = [0]↤[[0]↤[[0]↤[3]↤[3]]↤[3]]↤[3] = [0]↤[[0]↤[[0]↤[2]↤[[0]↤[2]↤[[0]↤[2]↤[3]]]]↤[3]]↤[3]
[1]↤[1]↤[a] = [1]↤[0]↤[[1]↤[0]↤[...[1]↤[0]↤[[1]↤[0]↤[a]]...]] with a nestings
[1]↤[b]↤[a] = [1]↤[b-1]↤[[1]↤[b-1]↤[...[1]↤[b-1]↤[[1]↤[b-1]↤[a]]...]] with a nestings
[c]↤[0]↤[a] = [c-1]↤[[c-1]↤[...[c-1]↤[[c-1]↤[a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[c]↤[b]↤[a] = [c]↤[b-1]↤[[c]↤[b-1]↤[...[c]↤[b-1]↤[[c]↤[b-1]↤[a]]...]] with a nestings
[3]↤[3]↤[3] = [3]↤[2]↤[[3]↤[2]↤[[3]↤[2]↤[3]]]
and this continues for any number of sets of [ ].
[4]↤[4]↤[4]↤[4] = [4]↤[4]↤[3]↤[[4]↤[4]↤[3]↤[[4]↤[4]↤[3]↤[[4]↤[4]↤[3]↤[4]]]]
= [4]↤[4]↤[3]↤[[4]↤[4]↤[3]↤[[4]↤[4]↤[3]↤[[4]↤[4]↤[2]↤[[4]↤[4]↤[2]↤[[4]↤[4]↤[2]↤[[4]↤[4]↤[2]↤[4]]]]]]]
[0, 0]↤[a] = [a]↤[a]↤[a]...[a]↤[a]↤[a] with a [a]s
[0, 0]↤[3] = [3]↤[3]↤[3]
[0, 0]↤[4] = [4]↤[4]↤[4]↤[4]
[0, 0]↤[[0, 0]↤[3]] = [0, 0]↤[[3]↤[3]↤[3]]
[0, b]↤[a] = [0, b-1]↤[[0, b-1]↤[...[0, b-1]↤[[0, b-1]↤[a]]...]] with a nestings
[c, 0]↤[a] = [c-1, [c-1, ...[c-1, [c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
if there is only 1 variable in the array (other than a or 0) the recursion happens at the closest 0 to the right.
[c, b]↤[a] = [c, b-1]↤[[c, b-1]↤[...[c, b-1]↤[[c, b-1]↤[a]]...]] with a nestings
[0, 0, 0]↤[a] = [[...[[a, a]↤[a], a]↤[a]..., a]↤[a], a]↤[a] with a nestings
[0, 0, 1]↤[a] = [0, 0, 0]↤[[0, 0, 0]↤[...[0, 0, 0]↤[[0, 0, 0]↤[a]]...]] with a nestings
[0, 0, b]↤[a] = [0, 0, b-1]↤[[0, 0, b-1]↤[...[0, 0, b-1]↤[[0, 0, b-1]↤[a]]...]] with a nestings
[0, c, 0]↤[a] = [0, c-1, [0, c-1, ...[0, c-1, [0, c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[d, 0, 0]↤[a] = [d-1, [d-1, ...[d-1, [d-1, a, 0]↤[a], 0]↤[a]..., 0]↤[a], 0]↤[a] with a nestings
[0, c, b]↤[a] = [0, c, b-1]↤[[0, c, b-1]↤[...[0, c, b-1]↤[[0, c, b-1]↤[a]]...]] with a nestings
[d, 0, b]↤[a] = [d, 0, b-1]↤[[d, 0, b-1]↤[...[d, 0, b-1]↤[[d, 0, b-1]↤[a]]...]] with a nestings
[d, c, 0]↤[a] = [d, c-1, 0]↤[[d, c-1, 0]↤[...[d, c-1, 0]↤[[d, c-1, 0]↤[a]]...]] with a nestings
[d, c, b]↤[a] = [d, c, b-1]↤[[d, c, b-1]↤[...[d, c, b-1]↤[[d, c, b-1]↤[a]]...]] with a nestings
and so on, with any number of entries in the [ ]
[0, 0]↤[0]↤[a] = [a, a, a...a, a, a]↤[a] with a entries of a
[0, 0]↤[b]↤[a] = [0, 0]↤[b-1]↤[[0, 0]↤[b-1]↤[...[0, 0]↤[b-1]↤[[0, 0]↤[b-1]↤[a]]...]] with a nestings
[0, 0]↤[0, 0]↤[a] = [[...[[a, a]↤[a]↤[a], a]↤[a]↤[a]..., a]↤[a]↤[a], a]↤[a]↤[a] wi
[0, 0]↤[0, b]↤[a] = [0, 0]↤[0, b-1]↤[[0, 0]↤[0, b-1]↤[...[0, 0]↤[0, b-1]↤[[0, 0]↤[0, b-1]↤[a]]...]] with a nestings
[0, 0]↤[c, 0]↤[a] = [0, 0]↤[c-1, [0, 0]↤[c-1, ...[0, 0]↤[c-1, [0, 0]↤[c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[0, 0]↤[c, b]↤[a] = [0, 0]↤[c, b-1]↤[[0, 0]↤[c, b-1]↤[...[0, 0]↤[c, b-1]↤[[0, 0]↤[c, b-1]↤[a]]...]] with a nestings
[0, d]↤[0, 0]↤[a] = [0, d-1]↤[0, 0]↤[[0, d-1]↤[0, 0]↤[...[0, d-1]↤[0, 0]↤[[0, d-1]↤[0, 0]↤[a]]...]] with a nestings
[0, d]↤[0, b]↤[a] = [0, d]↤[0, b-1]↤[[0, d]↤[0, b-1]↤[...[0, d]↤[0, b-1]↤[[0, d]↤[0, b-1]↤[a]]...]] with a nestings
[0, d]↤[c, 0]↤[a] = [0, d]↤[c-1, [0, d]↤[c-1, ...[0, d]↤[c-1, [0, d]↤[c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[0, d]↤[c, b]↤[a] = [0, d]↤[c, b-1]↤[[0, d]↤[c, b-1]↤[...[0, d]↤[c, b-1]↤[[0, d]↤[c, b-1]↤[a]]...]] with a nestings
[e, 0]↤[0, 0]↤[a] = [e-1, [e-1, ...[e-1, [e-1, a]↤[0, 0]↤[a]]↤[0, 0]↤[a]...]↤[0, 0]↤[a]]↤[0, 0]↤[a] with a nestings
[e, 0]↤[0, b]↤[a] = [e, 0]↤[0, b-1]↤[[e, 0]↤[0, b-1]↤[...[e, 0]↤[0, b-1]↤[[e, 0]↤[0, b-1]↤[a]]...]] with a nestings
[e, 0]↤[c, 0]↤[a] = [e, 0]↤[c-1, [e, 0]↤[c-1, ...[e, 0]↤[c-1, [e, 0]↤[c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[e, 0]↤[c, b]↤[a] = [e, 0]↤[c, b-1]↤[[e, 0]↤[c, b-1]↤[...[e, 0]↤[c, b-1]↤[[e, 0]↤[c, b-1]↤[a]]...]] with a nestings
[e, d]↤[0, 0]↤[a] = [e, d-1]↤[0, 0]↤[[e, d-1]↤[0, 0]↤[...[e, d-1]↤[0, 0]↤[[e, d-1]↤[0, 0]↤[a]]...]] with a nestings
[e, d]↤[0, b]↤[a] = [e, d]↤[0, b-1]↤[[e, d]↤[0, b-1]↤[...[e, d]↤[0, b-1]↤[[e, d]↤[0, b-1]↤[a]]...]] with a nestings
[e, d]↤[c, 0]↤[a] = [e, d]↤[c-1, [e, d]↤[c-1, ...[e, d]↤[c-1, [e, d]↤[c-1, a]↤[a]]↤[a]...]↤[a]]↤[a] with a nestings
[e, d]↤[c, b]↤[a] = [e, d]↤[c, b-1]↤[[e, d]↤[c, b-1]↤[...[e, d]↤[c, b-1]↤[[e, d]↤[c, b-1]↤[a]]...]] with a nestings
and so on, with any number of arrays with any number of entries in them.
[0]↤↤[a] = [a, a, a...]↤[a, a, a...]↤......↤[a, a, a...]↤[a, a, a...]↤[a] with a [a, a, a...] with a a's
[b]↤↤[a] = [b-1]↤↤[[b-1]↤↤[...[b-1]↤↤[[b-1]↤↤[a]]...]] with a nestings
[2]↤↤[2] = [1]↤↤[[1]↤↤[2]] = [1]↤↤[[0]↤↤[[0]↤↤[2]]] = [1]↤↤[[0]↤↤[[2, 2]↤[2, 2]↤[2]]]
this continues in the same way as with 1 ↤, except with 2.
There can be any number of arrows.
[0]↤↤↤[3] = [3, 3, 3]↤↤[3, 3, 3]↤↤[3, 3, 3]↤[3]
[1]↤↤↤[3] = [0]↤↤↤[[0]↤↤↤[[0]↤↤↤[3]]] = [0]↤↤↤[[0]↤↤↤[[3, 3, 3]↤↤[3, 3, 3]↤↤[3, 3, 3]]]
[0]↤↤↤↤[5] = [5, 5, 5, 5, 5]↤↤↤[5, 5, 5, 5, 5]↤↤↤[5, 5, 5, 5, 5]↤↤↤[5, 5, 5, 5, 5]↤↤↤[5, 5, 5, 5, 5]↤↤↤[5]
[2]↤↤↤↤↤[3] = [1]↤↤↤↤↤[[1]↤↤↤↤↤[[1]↤↤↤↤↤[3]]]
[b]↤c[a] = [b]↤↤...↤↤[a] with c arrows
(This extension is not necessarily fully integrated, but is still treated as such.)
[0]2↤[a] = [a]↤a[a]
[0]2↤[3] = [3]↤↤↤[3]
[b]2↤[a] = [b-1]2↤[[b-1]2↤[...[b-1]2↤[[b-1]2↤[a]]...]] with a nestings
[1]2↤[2] = [0]2↤[[0]2↤[2] = [0]2↤[[2]↤↤[2]] = [0]2↤[[1]↤↤[[0]↤↤[[2, 2]↤[2, 2]↤[2]]]]
[0]2↤[0]2↤[a] = [[...[[a]2↤[a]]2↤[a]...]2↤[a]]2↤[a] with a nestings
[0]2↤[0]2↤[3] = [[[3]2↤[3]]2↤[3]]2↤[3] = [[[2]2↤[[2]2↤[[2]2↤[3]]]]2↤[3]]2↤[3] = [[[2]2↤[[2]2↤[[1]2↤[[1]2↤[[1]2↤[3]]]]]]2↤[3]]2↤[3]
= [[[2]2↤[[2]2↤[[1]2↤[[1]2↤[[0]2↤[[0]2↤[[0]2↤[3]]]]]]]]2↤[3]]2↤[3] = [[[2]2↤[[2]2↤[[1]2↤[[1]2↤[[0]2↤[[0]2↤[[3]↤↤↤[3]]]]]]]]2↤[3]]2↤[3]
a
Firstly, a = ||...|| with a |'s
1 = |, 2 = ||, 3 = |||, 4 = ||||, and so on.
°a = a
a° = -a
°||| = -3
Note that any more than ||||| can be shorthanded as the actual value. For example: ||||||| can be shortened to 7.
a~ = change applied to a = a+1
a~b = a+b = a changed by b
||~|| = 2 changed by 2 = |||| = 4, |||~||||| = 3 changed by 5 = 8
a~b~c works right to left. ||~|||~||||| = ||~|||||||| = 11
a~~b = a~a~a...a~a~a with b a's = b changes of a
a~~ = a~
|||~~||| = |||~|||~||| = 9, ||~~||||| = 10
a~~~b = a~~a~~a... with b a's = a twice changed by b
a~~~ = a~~ = a~
|||~~~||| = |||~~|||~~||| = |||~~||||||||| = 3*9 = 27
and so on, with any number of ~'s
a~^(c)b = a~~..(c)..~~b
..(c).. means with c recursions, or in this case c ~'s
Functions can be defined within a string. Special built-in symbols are used. Symbols include: W = where, IF = if, TH = then, E = else, EIF = else if, T = true, F = false, A = and, OR = or, N = not, XOR = exclusive or, WHL = while loop, FOR = for loop, ABS = absolute value, CM = Case matching (replaces nested ifs. CMa:(3:F,4:T,E:Y) This means if a is 3, then do F, if a is 4, do T, else to Y. CMa(1:o,>x:f) also works for if a is 1, do o, if a is greater than x, do f). : is generally similar to =, N: means not equal =. Parenthesis *can* be used to clarify. < and > can also be used. Then symbols don't have to be used after ifs, but can to clarify. ; at the end of a sequence concludes it so multiple lines can share the same logic. Arrays can be made using any symbols. Like so: k:{a, b, c} means thee variable "k" holds the array {a, b, c}. LEN(k) then produces the amount of the entries in k, in this case 3.
Functions are used like so:
|||H||,WaHb:a~^(a~^(..(b)..a~^(a)b..(b)..)b)b
In the second part, WaHb:...(definition), the WaHb means where aHb:... which means when a is before and b is after, the function is applied. This particular function outputs a~^(a~^(..(b)..a~^(a)b..(b)..)b)b. As stated earlier, ..(c).., or in this case ..(b).., means with b repetitions. At least 2 repetitions are needed to be written out to define the pattern. So this whole statement, |||H||,WaHb:a~^(a~^(..(b)..a~^(a)b..(b)..)b)b, turns out to be |||~^(|||~^(||)||)||. There are 2 nestings, with the "b" taking the inner most input, as stated in H's definition. |||~^(|||~^(||)||)|| = |||~^(|||~~||)|| = |||~^(||||||)|| = |||~~~~~~||, or 3 octated to 2. If a negative input is given into a function that only works with positive inputs, just use the absolute value of the number, unless otherwise stated.
Another example:
@||,°|||,W(@a,b:IF(b>0):a~~~b,E:a~b)
In this example, the function @ is used and defined. It is defined as: if b is positive, then the output is a~~~b, if b is negative, the output is a~b. In this example, b is negative, so @||,°|||,W(@a,b:IF(b>0):a~~~b,E:a~b) = ||~°||| = -1
Functions can be defined and/or used within other functions
&a,W&a:%(..(a)..%(%a)..(a)..),W%a:a~~..(a)..~~a
In this example, the function % is defined within the function &. If we took &|||,W&a:%(..(a)..%(%a)..(a)..),W%a:a~~..(a)..~~a = %%%||| = %%(|||~~~|||) = %%(|||||||||||||||||||||||||||) = %%27 = %(27~~..(27)..~~27), which is a *very* big number.
Any function can be described and used in these strings.
Triangle Hexagon Ordinal Matrix Notation
# can be any array or an empty array. Definitions with "1"s take priority. "ω" is not a variable.
◭a◮ = a+1
◭a, 1◮ = a*2 (= ◭◭...◭◭a◮◮...◮◮ with a nestings)
◭a, 2◮ = a*2^a
◭a, b, #◮ = ◭◭...◭◭a, b-1, #◮, b-1, #◮..., b-1, #◮, b-1, #◮ with a nestings
◭3, 2◮ = ◭◭◭3, 1◮, 1◮, 1◮ = ◭◭6, 1◮, 1◮ = ◭12, 1◮ = 24
◭3, 3◮ = ◭◭◭3, 2◮, 2◮, 2◮ = ◭◭24, 2◮, 2◮ = ◭◭24, 2◮, 2◮ =◭402,653,184, 2◮ = (402,653,184)*2^(402,653,184)
◭a, 1, 1, 1...(b 1's)...1, 1, 1◮ = ◭a, a, a...(b-1 a's)...a, a, ◭a, a, a...(b-1 a's)...a, a, ...◭a, a, a...(b-1 a's)...a, a, ◭a, a, a...(b a's)...a, a, a◮◮...◮◮ with a nestings
◭3, 1, 1, 1◮ = ◭3, 3, ◭3, 3, ◭3, 3, 3◮◮◮
◭4, 1, 1, 1, 1, 1◮ = ◭4, 4, 4, 4, ◭4, 4, 4, 4, ◭4, 4, 4, 4, ◭4, 4, 4, 4, 4◮◮◮◮
◭a, 1, 1, 1...(b 1's)...1, 1, 1, c, #◮ = ◭a, a, a...(b a's)...a, a, ◭a, a, a...(b a's)...a, a, ...◭a, a, a...(b+1 a's)...a, a, a, c-1, #◮..., c-1, #◮, c-1, #◮ with a nestings
◭3, 1, 1, 2◮ = ◭3, 3, ◭3, 3, ◭3, 3, 3, 1◮, 1◮, 1◮
◭3, 1, 1, 1, 3◮ = ◭3, 3, 3, ◭3, 3, 3, ◭3, 3, 3, 3, 2◮, 2◮, 2◮
◭4, 1, 1, 3, 2◮ = ◭4, 4, ◭4, 4, ◭4, 4, ◭4, 4, 4, 2, 2◮, 2, 2◮, 2, 2◮, 2, 2◮
a⎔1 = ◭a, a, a...a, a, a◮ with a a's
a⎔b = ((...(a⎔b-1)⎔b-1...)⎔b-1)⎔b-1 with a nestings
a⎔ω = a⎔a
a⎔ω+1 = ((...(a⎔ω)⎔ω...)⎔ω)⎔ω with a nestings
3⎔ω+1 = ((3⎔ω)⎔ω)⎔ω = ((((3⎔2)⎔2)⎔2)⎔ω)⎔ω = ((((3⎔2)⎔2)⎔2)⎔ω)⎔ω = ((((((3⎔1)⎔1)⎔1)⎔2)⎔2)⎔ω)⎔ω = (((((◭3, 3, 3◮⎔1)⎔1)⎔2)⎔2)⎔ω)⎔ω
a⎔ω+b+1 = ((...(a⎔ω+b)⎔ω+b...)⎔ω+b)⎔ω+b
3⎔ω+2 = ((3⎔ω+1)⎔ω+1)⎔ω+1
3⎔ω+3 = ((3⎔ω+2)⎔ω+2)⎔ω+2
a⎔ω+ω = a⎔ω2
3⎔ω+ω = ((3⎔ω+2)⎔ω+2)⎔ω+2
4⎔ω2 = (((4⎔ω+3)⎔ω+3)⎔ω+3)⎔ω+3
a⎔ωb = a⎔ω*b
3⎔ω3 = 3⎔ω*3 = 3⎔ω+ω+ω = ((3⎔ω+ω+2)⎔ω+ω+2)⎔ω+ω+2 = ((((3⎔ω+ω+1)⎔ω+ω+1)⎔ω+ω+1)⎔ω+ω+2)⎔ω+ω+2
= ((((((3⎔ω+ω)⎔ω+ω)⎔ω+ω)⎔ω+ω+1)⎔ω+ω+1)⎔ω+ω+2)⎔ω+ω+2
= ((((((((3⎔ω+2)⎔ω+2)⎔ω+2)⎔ω+ω)⎔ω+ω)⎔ω+ω+1)⎔ω+ω+1)⎔ω+ω+2)⎔ω+ω+2
...
3⎔ω4 = 3⎔ω+ω+ω+ω = ((3⎔ω+ω+ω+3)⎔ω+ω+ω+3)⎔ω+ω+ω+3 = ((3⎔3ω+3)⎔3ω+3)⎔3ω+3
a⎔ω*ω = a⎔ω^2
3⎔ω^2 = 3⎔ω*ω = 3⎔ω3
4⎔ω^2 = 4⎔ω*ω = 4⎔ω4
a⎔ω^b = a⎔ω*ω*ω...ω*ω with b ω's
3⎔ω^3 = 3⎔ω*ω*ω = ((3⎔ω*ω*2)⎔ω*ω*2)⎔ω*ω*2
a⎔ω^ω = a⎔ω^^2 = a⎔ω*ω*ω...ω*ω with a ω's
3⎔ω^ω = 3⎔ω^3 = 3⎔ω*ω*ω
4⎔ω^ω = 4⎔ω^4 = 4⎔ω*ω*ω*ω
and so on, ω being able to be modified by any operator.
For example 3⎔ω^^^ω = 3⎔ω^^ω^^ω = 3⎔ω^^ω^^3 ...
things like a⎔◭ω, #◮ or even a⎔ω⎔ω are possible, which is simply to use the operator ⎔ω on ω.
3⎔◭ω, 1◮ for example, is just 3⎔ω2
Something like 3⎔3⎔3 would be evaluated like 3⎔(3⎔3), and 3⎔3⎔3⎔3 would be 3⎔(3⎔(3⎔3))
a⎔⎔1 = a⎔ω⎔ω...⎔ω⎔ω with a ω's
a⎔⎔b = ((...(a⎔⎔b-1)⎔⎔b-1...)⎔⎔b-1)⎔⎔b-1with a nestings
a⎔⎔ω = a⎔⎔a
a⎔⎔ω+1 = ((...(a⎔⎔ω)⎔⎔ω...)⎔⎔ω)⎔⎔ω with a nestings
This continues in the same way as a⎔ω. It can also be modified with any operator. In fact, a⎔⎔...(b ⎔'s)...⎔⎔1 = a⎔⎔...(b-1)..⎔⎔ω⎔⎔...(b-1)..⎔⎔ω...(a)...⎔⎔...(b-1)..⎔⎔ω⎔⎔...(b-1)..⎔⎔⎔ω. This means a⎔⎔...(b ⎔'s)...⎔⎔c = ((...(((a)⎔⎔...(b ⎔'s)...⎔⎔c-1)⎔⎔...(b ⎔'s)...⎔⎔c-1)...)⎔⎔...(b ⎔'s)...⎔⎔c-1)⎔⎔...(b ⎔'s)...⎔⎔c-1 with a nestings. Mixed things like a⎔⎔ω⎔ω⎔⎔ω⎔⎔⎔ω⎔ω are possible, due to ⎔⎔...⎔⎔ω being an operator.
a[⎔]1 = a⎔⎔...(a)...⎔⎔a
a[⎔]b = ((...(a[⎔]b-1)[⎔]b-1...)[⎔]b-1)[⎔]b-1 with a nestings
a[⎔]ω = a[⎔]a
The ω on this function can also be modified with operators. This means things like a[⎔]ω[⎔]ω[⎔]ω are possible, and further. This naturally grows faster than most hierarchies. It can also be expanded to a[⎔⎔]ω or a[⎔⎔⎔]ω and further.
a[⎔k]b = a[⎔⎔⎔...(k)...⎔⎔⎔]b
a[⎔k]1 = a[⎔k-1]ω[⎔k-1]ω...ω[⎔k-1]ω[⎔k-1]ω with a ω's
a[⎔k]b = ((...(a[⎔k]b-1)[⎔k]b-1...)[⎔k]b-1)[⎔k]b-1 with a nestings
a[⎔k]ω = a[⎔k]a
ω can be modified in any way.
a[⎔k]ω+1 = ((...(a[⎔k]ω)[⎔k]ω...)[⎔k]ω)[⎔k]ω with a nestings
a[⎔k]ω+b = ((...(a[⎔k]ω+(b-1))[⎔k]ω+(b-1)...)[⎔k]ω+(b-1))[⎔k]ω+(b-1) with a nestings
a◈kb = a◈◈◈...(k)...◈◈◈b
a◈10 = a[⎔a]ω[⎔a]ω...ω[⎔a]ω[⎔a]ω with a ω's
a◈11 = ((...(a◈0)◈0...)◈0)◈0 with a nestings
a◈k1 = a◈k-1ω◈k-1ω...ω◈k-1ω◈k-1ω with a ω's
a◈kb = ((...(a◈kb-1)◈kb-1...)◈kb-1)◈b-1 with a nestings
a◈kω = a◈ka
again, ω can be modified in any way.
a◈kω+1 = ((...(a◈kω)◈kω...)◈kω)◈kω with a nestings
a◈kω+b = ((....(a◈kω+(b-1))◈kω+(b-1)...)◈kω+(b-1))◈kω+(b-1) with a nestings
a[◈]1 = a◈aω◈aω...ω◈aω◈aω with a ω's
a[◈]b = ((...(a[◈]b-1)[◈]b-1...)[◈]b-1)[◈]b-1 with a nestings
a[◈]ω = a[◈]a
ω can be modified in any way.
a[◈]ω+1 = ((...(a[◈]ω)[◈]ω...)[◈]ω)[◈]ω with a nestings
a[◈]ω+b = ((...(a[◈]ω+(b-1))[◈]ω+(b-1)...)[◈]ω+(b-1))[◈]ω+(b-1) with a nestings
The name is complete nonsense. I just wanted a cool acronym, which is OTHORN.
n#0 = n+1
n#a = ((...(n#a-1)#a-1...)#a-1)#a-1 with n nestingsω
n#ω = n#n
ω can be modified with operators. For example n#ω+1 = ((...(n#ω)#ω...)#ω)#ω with n nestings, and n#ω3 = ((...(n#ω+ω+(n-1))#ω+ω+(n-1)...)#ω+ω+(n-1))#ω+ω+(n-1)
n#ωω = n#(ω#ω)
ωω can also be modified. In fact, any stack of ω can be modified.
n##...(g #'s)...##ωω..(k ω's)..ωω = n##...(g #'s)...##(ω#ωω..(k-1 ω's)...ωω)
n##...(g #'s)...##0 = n##...(g-1 #'s)...##)ωω..(n ω's)...ωω
n##...(g #'s)...##a = ((...(n##...(g #'s)...##a-1)##...(g #'s)...##a-1...)##...(g #'s)...##a-1)##...(g #'s)...##a-1
n##...(g #'s)...##ω = n##...(g #'s)...##n
again, ω can be modified no matter context.
n#Ω = n##...(n)...##ωω...(n)...ωω
n#Ω+1 = ((...(n#Ω)#Ω...)#Ω)#Ω with n nestings
Ω can be modified in any way.
n##...(g #'s)...##ΩΩ..(k Ω's)..ΩΩ = n##...(g #'s)...##(Ω#ΩΩ..(k-1 Ω's)...ΩΩ)
n#[1] = n##...(n #'s)...##ΩΩ...(n Ω's)...ΩΩ
n#[a] = ((...(n#[a-1])#[a-1]...)#[a-1])#[a-1] with n nestings
n#[ω] = n#[n]
ω can be modified with operators
n#[ω+1] = ((...(n#[ω])#[ω]...)#[ω])#[ω]
where # is any or empty array
<a> = fω(a)
<a, 0> = <<...<<a>>...>> with a nestings
<a, b, #> = <<...<<a, b-1, #>, b-1, #>..., b-1, #>, b-1, #>> with a nestings
<a, 0, 0, 0...(b 0's)...0, 0, 0> = <a, a, a...(b-1 a's)...a, a, <a, a, a...(b-1 a's)...a, a, ...<a, a, a...(b-1 a's)...a, a, <a, a, a...(b a's)...a, a, a>>...>> with a nestings
<a, 0, 0, 0...(b 0's)...0, 0, 0, c, #> = <a, a, a...(b a's)...a, a, <a, a, a...(b a's)...a, a, ...<a, a, a...(b+1 a's)...a, a, a, c-1, #>..., c-1, #>, c-1, #> with a nestings
<a>[0] = <a, a, a...a, a, a> with a a's
<a>[k] = <<...<<a>[k-1]>[k-1]...>[k-1]>[k-1] with a nestings
<a, 0>[0] = <a>[<a>[...<a>[<a>[a]]...]] with a nestings
<3, 0>[0] = <3>[<3>[<3>[3]]]
<a, b, #>[0] = <<...<<a, b-1, #>[0], b-1, #>[0]..., b-1, #>[0], b-1, #>[0] with a nestings
<3, 1>[0] = <<<3, 0>[0], 0>[0], 0>[0]
a
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