The Strong Horseshoe Notation is a notation relative to amaller numbers instead of bigger ones. This notation takes inspiration from the Ackermann function and John Bower's BEAF, along with O
DESCRIPTION:
n^-1 = 1 / n
{x,x,z} = x^...(x arrows)...^x
{x,x,x,x} = {x,{x,x-1,x,x},x-1,x} (if x > 1)
x&y = {x,x,x,(y entries),x,x,x}
{{{{{{{{{{n}}}}}}}}}} = {n}¹⁰
Ʊ[n] (If n>1) = (n&(Ʊ[n-1]^-1)) {{{...n-1 brackets...{{{n}}}...n-1 brackets...}}} (-n)
Ʊ[0] = 1
Ʊ[1] = 0.1
Ʊ[Ʊ[n]] (If n>1) = Ʊ[(n&(Ʊ[n-1]^-1)) {{{...n-1 brackets...{{{n}}}...n-1 brackets...}}} (n)]
(if there is an even amount of horseshoe functions, then the alternative at the end is changed to (n), since a decimal^-n is a larger number (e.g. 0.5^-2 = 4.) If it's an odd amount, then it isn't changed.)
EXAMPLES:
Ʊ[0] = 1
Ʊ[1] = 0.1
Ʊ[2] = (2&10){2}(-2) since 2&10 = {2,2,2,2,2,2,2,2,2,2} & Ʊ[1]^-1 = 10
Ʊ[3] = 3&((3&10){2}(2)) {{3}} (-3)
Ʊ[10] = (10&(Ʊ[9]^-1)) {8}⁹ (-10)
The Weak Horseshoe Notation is a Notation which is a relative to the Strong Horseshoe Notation, instead changing the function to be bigger than an SHN value. This function is both inspired from the Ackermann function and Knuth's Arrow Notation.
DESCRIPTION:
n^-1 = 1 / n
ʊ[n] (If n>1) = (n↑...((ʊ[n-1]^-1) arrows)...↑n)^(-1)
ʊ[0] = 1
ʊ[1] = 0.1
ʊ[ʊ[n]] (If n>1) = ʊ[(n↑...((ʊ[n-1]^-1) arrows)...↑n)^(1)]
(if there is an even amount of horseshoe functions, then the alternative at the end is changed to (1), since a decimal^-1 is a larger number (e.g. 0.5^-1 = 2.) If it's an odd amount, then it isn't changed.)
EXAMPLES:
ʊ[0] = 1
ʊ[1] = 0.1
ʊ[2] = (2↑↑↑↑↑↑↑↑↑↑2)^-1)
ʊ[3] = (3↑...((ʊ[2]^-1) arrows)...↑3)^-1)
ʊ[10] = (10↑...((ʊ[9]^-1) arrows)...↑10)^-1)
Thereforal function takes inspiration from BEAF + Knuth Arrow Notation, and makes the function of growth smaller instead of bigger.
Description:
{a∴b∴c} = a+(1/(b^c))
{a∴b∴c∴d} = a+(1/(b↑...(d arrows)...↑c))
{a∴b∴c∴d∴e} = a+(1/(b↑...((d↑...(e arrows...)...↑d) arrows)...↑c))
{a∴b∴c∴d∴e∴f} = a+(1/(b↑...((d↑...((e↑...(f arrows...)...↑e) arrows...)...↑d) arrows)...↑c))
Any varaibles after a,b and c, stack the amount of arrows between b and c. (As seen above with d,e and f.)
{a∴b∴c∴d∵e} = a+(1/(b↑...(d↑...(e stacks of d arrows)...↑d arrows)...↑c))
EXAMPLES:
{1∴10∴0} = 2 (1/1 = 1)
{1∴2∴1} = 1.5
{1∴10∴2} = 1.001
{5∴3∴3} = 5.037037037...
{10∴10∴10} = 10.0000000001
{10∴10∴10∴10} = 10 + (1/10↑↑↑↑↑↑↑↑↑↑10)
{10∴10∴10∴10∴10∴10} = 10 + (1/10↑...(10↑...(10 arrows)...↑10))...↑10)
{10∴2∴2∴2∵2} = 10+(1/2↑...(2↑↑↑↑2 arrows)...↑2) = 10.25
{10∴2∴2∴2∵3} = 10+(1/3↑...(3↑...(3↑↑↑↑↑3 arrows)...↑3 arrows)...↑3)
This function, of course, takes inspiration from the standard of naming -illions, and makes them simpler to write out. It is also similar to the Zillion function, but this function expands on it by including other factors.
DESCRIPTION:
IS[a,0,c] = a+c
IS[a,1,c] = a^c
IS[a,b,c] = a^(3x10^(...b times...(3x10^c))+3) if b ≥ 1
EXAMPLES:
IS[2,1,2] = 4 (2^2)
IS[10,1,33] = 10^33 = Decillion
IS[10,2,33] = 10^(3x10^33)+3 = Mecillion
IS[10,3,30] = 10^((3x10^)3x10^30)+3 = Quettillion
IS[10,5,30] = 10^((3x10^)3x10^)3x10^)3x10^30)+3 = Geopillion
IS[10,10,30] = 10^((3x10^)3x10^)3x10^)3x10^)3x10^)3x10^)3x10^)3x10^)3x10^30)+3 = Teethillion
EXTENSION:
IS[a,1,c,1] = a^c
IS[a,b,c,1] = IS[a,b,c]
IS[a,b,c,2] = IS[a,IS[a,b,c],c]
IS[a,b,c,3] = IS[a,IS[a,IS[a,b,c],c],c]
IS[a,b,c,d] = IS[a,IS[a,IS[a,...[d IS[a,b,c] chunks...],c],c],c]
IS[a,b,c,d,e] = IS[a,b,c,IS[a,b,c,IS[a,b,c,...e IS[a,b,c,d] chunks...]]]
IS[a,b,c,d,e,f] = IS[a,b,c,d,IS[a,b,c,d,IS[a,b,c,d,...f IS[a,b,c,d,e,] chunks...]]]
IS[a,b,c,d,e,f,g] = IS[a,b,c,d,e,IS[a,b,c,d,e,IS[a,b,c,d,e,...g IS[a,b,c,d,e,f,] chunks...]]]
IS[0#0] = IS[0,0,0] = 0+0 = 0
IS[1#0] = IS[1,1,1] = 1^1 = 1
IS[2#0] = IS[2,2,2] = 2^(3x10^2)+3 = 2^303 = 1.62962...x10^91
IS[3#0] = IS[3,3,3] = 3^((3x10^)3x10^3)+3 = 3^(3x10^303)
IS[5#0] = IS[5,5,5,5,5]
IS[n#0] = IS[n,n,n,n,n,n,n... (n list of variables)] if n ≥ 3
IS[0#1] = IS[n#0]
IS[1#1] = IS[IS[n#0]#0]
IS[2#1] = IS[IS[IS[n#0]#0]#0]
IS[0#2] = IS[n#1] = IS[IS[IS[...(n+1 IS[n#0] stacks...)#0]#0]#0]
IS[0#n] = IS[n#n-1]
The WALKER function is a function created by BraidenNO10 (Aka The creator of this website and its pages.) It takes inspiration from Knuth's Arrow Notation And the Ackermann Function. This one is the most expansive computable function I've created .
DESCRIPTION (Copied from my oldest Googology blog post)
W[0,n] = n
W[1,n] = n↑n = nn = n↑0n
W[2,n] = n↑...(n arrows)...↑n = n↑1n
n↑...(n↑1n arrows)...↑n = n↑1↑1n
n↑...(n↑1↑1n arrows)...↑n = n↑1↑1↑1n
A = B-1
n↑...(n(A of ↑1)n of arrows)...↑n = n(B ↑1s)n
n↑1...(n of ↑1s)...↑1n = n↑2n
W[(m+2),n] = n↑m...(n of ↑ms)...↑mn = n↑m+1n
n↑m...(n↑m+1n of ↑m)...↑mn = n↑m+1↑m+1n
n↑m...(n↑m+1↑m+1n of ↑m)...↑mn = n↑m+1↑m+1↑m+1n
A = B-1
n↑m...(n(A of ↑m+1)n of ↑ms)...↑mn = n(B ↑m+1s)n
W[(m+3),n] = n↑m+1...(n of ↑m+1s)...↑m+1n = n↑m+2n
W[(m if >1),n] = n↑m-1n
W[1,n] = W1[n] = W[1-n]
W1[W1[W1...(W1[n] W1s)...[W1[W1[W1[n]]]...]]] = W2[n]
W2[W2[W2...(W2[n] W2s)...[W2[W2[W2[n]]]...]]] = W3[n] and so on.
This results: Wm[Wm[Wm...(Wm[n] Wms)...[Wm[Wm[Wm[n]]]...]]] = Wm+1[n]
EXTENSION:
WW[n][n] = W[2-n]
WW[n]W[n][n] = W[3-n]
WW[n]...(W[n] W[n]s)[n] = W[n-n]
WW[n-n]...(W[n-n] W[n-n]s)[n-n] = W[n-n-n]
o = n entries of n
WW[o]...(W[o] W[o]s)[n-1 entries of n] = W[n entries of n] = W[n--n]
WW[o--o]...(W[o--o] W[o--o]s)[n--n] =W[n---n] and so on
WW[o-(n dashes)-o]...(W[o-(n dashes)-o] W[o-(n dashes)-o]s)[n-(n dashes)-n] =W[n-(n+1dashes)-n]
W(W[o-(W[n-n] dashes)-o] W[o-(W[n-n] dashes)-o]s)[n-(W[n-n] dashes)-n] =W[n/1n]
W(W[/1-(W[n/1n] slashes)/1o] W[o/1(W[n/1n] slashes)/1o]s)[n/(W[n/1n] slashes)/n] =W[n/1/1n]
W(W[/1/1-(W[n/1/1n] slashes)/1/1o] W[o/1/1(W[n/1/1n] slashes)/1/1o]s)[n/(W[n/1n] slashes)/n] =W[n/1/1/1n] and so on
W(W[/1/1-(W[n/1([W[n] /1s)/1n] slashes)/1/1o] W[o/1/1(W[n/1([W[n] /1s)/1n] slashes)/1([W[n] /1s)/1o]s)[n/(W[n/1n] slashes)/n] =W[n/1(n of /1)/1n]
W[n/1/1/1...(W[n/1n] slashes).../1/1/1n] = W[n/2n] (The slash rule from above still applies to /n of higher value.)
W[n/n/n/n...(W[n/nn] slashes).../n/n/nn] = W[n/n+1n]
W[n{0/2}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{0/1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{0/3}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{0/2}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{0/m}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{0/m-1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{0/n}n] / W[n{1/n}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{0/n-1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{2/n}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{2/n-1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{m/n}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{m/n-1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{n/n}n] / W[n{n/n}{0/0}n] = W[n/W{n/n]n/W{n/n]n/W{n/n]...(With W[n{n/n-1}n] sets of /s)...n/W{n/n]n/W{n/n]n/W{n/n]n]
W[n{n/n}{0/1}n] = W[n{n/W[n{n/n}{0/0}n]}{0/0}n]
W[n{n/n}{0/2}n] = W[n{n/W[n{n/W[n{n/n}{0/0}n]}{0/0}n]}{0/0}n]
W[n{n/n}{0/m}n] = W[n{n/W[n{n/W[n{n/...(m W sets)...{0/0}n]}{0/0}n]}{0/0}n]
W[n{n/n}{0/n}n] / W[n{n/n}{1/0}n] = W[n{n/W[n{n/W[n{n/...(n W sets)...{0/0}n]}{0/0}n]}{0/0}n]
W[n{n/n}{1/1}n] = W[n{n/W[n{n/n}{1/0}n]}{1/0}n]
W[n{n/n}{1/2}n] = W[n{n/W[n{n/W[n{n/n}{1/0}n]}{1/0}n]}{1/0}n]
W[n{n/n}{1/m}n] = W[n{n/W[n{n/W[n{n/...(m W sets)...{1/0}n]}{1/0}n]}{1/0}n]
W[n{n/n}{1/n}n] / W[n{n/n}{2/0}n] = W[n{n/W[n{n/W[n{n/...(n W sets)...{1/0}n]}{1/0}n]}{1/0}n]
W[n{n/n}{m/0}n] = W[n{n/W[n{n/W[n{n/...(n W sets)...{m-1/0}n]}{m-1/0}n]}{m-1/0}n]
W[n{n/n}{n/0}n] = W[n{n/W[n{n/W[n{n/...(n W sets)...{n-1/0}n]}{n-1/0}n]}{n-1/0}n]
W[n{n/n}{n/n}n] = W[n{n/n}{n/n}{0/0}n]
W[n{n/n}{n/n}{0/1}n] = W[n{n/W[n{n/n}{n/n}{0/0}n]}{n/W[n{n/n}{n/n}{0/0}n]}}{0/0}n]
W[n{n/n}{n/n}{0/n}n] / W[n{n/n}{n/n}{1/0}n] = W[n{n/W[n{n/...(n W sets)...}{n/...(n W sets)...}{0/0}n]}{n/W[n{n/...(n W sets)...}{n/...(n W sets)...}{0/0}n]}}{0/0}n]
W[n{n/n}{n/n}{1/1}n] = W[n{n/W[n{n/n}{n/n}{1/0}n]}{n/W[n{n/n}{n/n}{1/0}n]}}{1/0}n]
W[n{n/n}{n/n}{1/n}n] / W[n{n/n}{n/n}{2/0}n] = W[n{n/W[n{n/...(n W sets)...}{n/...(n W sets)...}{1/0}n]}{n/W[n{n/...(n W sets)...}{n/...(n W sets)...}{1/0}n]}}{1/0}n]
W[n{n/n}{n/n}{n/n}n] = W[n{0//3}n]
W[n{0//n}n] / W[n{1//0}n] = W[n{n/n}{n/n}{n/n}...(n {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{1//1}n] = W[n{n/n}{n/n}{n/n}...(W[n{1//0}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{1//m}n] = W[n{n/n}{n/n}{n/n}...(W[n{1//m-1}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{1//n}n] / W[n{2//0}n] = W[n{n/n}{n/n}{n/n}...(W[n{1//n-1}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{3//0}n] = W[n{n/n}{n/n}{n/n}...(W[n{2//n-1}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{m//0}n] = W[n{n/n}{n/n}{n/n}...(W[n{m-1//n-1}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{n//0}n] = W[n{n/n}{n/n}{n/n}...(W[n{n-1//n-1}n] {n/n}s)...{n/n}{n/n}{n/n}n]
W[n{n//n}n] = W[n{0///0}n]
W[n{n///n}n] = W[n{0////0}n]
W[n{n/...(n /s).../n}n] = W[n{0\0}n] / W[n(0(2)0)n]
W[n{0\1}n] = W[n{n/...(W[n{0\0}n] /s).../n}n]
W[n{0\n}n] / W[n{1\0}n] = W[n{n/...(W[n{0\n-1}n] /s).../n}n]
W[n{1\n}n] / W[n{2\0}n] = W[n{n/...(W[n{1\n-1}n] /s).../n}n]
W[n{m\0}n] = W[n{n/...(W[n{m-1\n-1}n] /s).../n}n]
W[n{n\0}n] = W[n{n/...(W[n{n-1\n-1}n] /s).../n}n]
W[n{n\n}n] = W[n{0\\0}n]
W[n{n\\n}n] = W[n{0\\\0}n]
W[n{n\\n}n] = W[n{0\\\0}n]
W[n{n\...(m \s)...\n}n] = W[n{n\...(m+1 \s)...\n}n]
W[n{n\...(n \s)...\n}n] / W[n(0(2)0)(0(2)0)...(n (0(2)0)s)...(0(2)0)(0(2)0)n] = W[n(0(3)0)n]
W[n(0(m)0)n] = W[n(0(m-1)0)(0(m-1)0)...(n (0(m-1)0)s)...(0(m-1)0)(0(m-1)0)n]
W[n(0(n)0)n] / W[n(0(0)1)n] = W[n(0(n-1)0)(0(n-1)0)...(n (0(n-1)0)s)...(0(n-1)0)(0(n-1)0)n]
W[n(0(1)1)n] = W[n(0(0)1)(0(0)1)...(n (0(0)1)s)...(0(0)1)(0(0)1)n]
W[n(0(n)1)n] = W[n(0(n-1)1)(0(n-1)1)...(n (0(n-1)1)s)...(0(n-1)1)(0(n-1)1)n]
W[n(0(n)n)n] / W[n(1(0)0)n] = W[n(0(n-1)n-1)(0(n-1)n-1)...(n (0(n-1)n-1)s)...(0(n-1)n-1)(0(n-1)n-1)n]
W[n(1(n)n)n] / W[n(2(0)0)n] = W[n(1(n-1)n-1)(1(n-1)n-1)...(n (1(n-1)n-1)s)...(1(n-1)n-1)(1(n-1)n-1)n]
W[n(3(0)0)n] = W[n(2(n)n)n]
W[n(m(0)0)n] = W[n(m-1(n)n)n]
W[n(n(0)0)n] = W[n(n-1(n)n)n]
EXAMPLES:
W[0,1] = 1
W[0,10] = 10
W[1,3] = 27
W[2,2] = 2↑↑2 = 4
W[2,3] = 3↑↑↑3 = 3↑↑↑↑2 = Tritri
W[2,10] = 10↑↑↑↑↑↑↑↑↑↑10 = {10,10,10} = Tridecal
W[3,2] = 2↑22
W[10,10] = 10↑92
Breaking down W[3,3]
3↑13 = 3↑↑↑3
3↑...(3↑↑↑3 arrows)...↑3= 3↑1↑13
3↑...(3↑...(3↑↑↑3 arrows)...↑3 arrows)...↑3 = 3↑1↑1↑13 = 3↑23
W[3,3] = 3↑...(3↑...(3↑↑↑3 arrows)...↑3 arrows)...↑3 ≈ {3,3,3,3,3}
This function is inspired by the Sam function, and it results in making bigger numbers, even larger than Sam's number, even if the real number entirely is ill defined. Therefore I'll be using Deeplinemadom's Sam Function in order to extend this. ( And use DLM's Sam function in TOS here.)
DESCRIPTION:
(Check DLM's Sam Function for Sam's Function used.)
TOS(-1) = Sam[10^100](10^100)
TOS(0) = Deeplinemadom's Sam's Number (aka. Sam[ω(ε(ω_1CK×f[ψ(K)×E100#100#100](Rayo(10^100))))CK](Large Number Garden Number))
TOS(1) = Sam[TOS(0)](TOS(-1))
TOS(2) = Sam[TOS(1)](TOS(0))
And so TOS(n) = Sam[TOS(n-1)](IS(n-2)), if n ≥ 1.
(No examples here srry)