T[n] = 3^n
T[1,2] = 3
so T[3] = 27, T[T[3]] = 7,625,597,484,987, T[T[T[3]]] = 3^^4
T^(n)[1] = 3^^n
T[2,2] = 3^^(T[1,2]) = 3^^3
T[3,2] = T^(T[2,2])[1] = 3^^3^^3
T[n,2] = T^(T[n-1,2])[1] = 3^^^n
T[2,3] = T[3,2]
T[3,3] = T[T[3,2],2]
T[4,3] = T[T[T[3,2],2],2]
2TN rules:
Base case: T[n,1] = 3^n
Degenerative case: T[1,n] = 1
Reduction case: T[2,n] = T[3,n-1]
Recursive case: T[a,b] = T[T[a-1,b],b-1]
Alternative Representations:
T[a,b] = {3,a,b} in BEAF/BAN.
T[a,b] = s(3,a,b) in SAN.
3TN rules:
Binary reduction: 3TN[a,b,1] = 2TN[a,b]
Degenerative case: T[1,b,c] = 3
Recursive case: T[a,b,c] = T[T[a-1,b,c],b-1,c]
Prime Reduction case: T[2,b,c] = T[3,b-1,c]
Full Reduction case: T[2,1,c] = T[3,3,c-1]
Alt. representation:
T[a,b,c] = {3,a,b,c} in BEAF/BAN
Full Ruleset:
Deleter case: T[a,b,c,...,x,y,z,1] = T[a,b,c,...,x,y,z]
Degenerative case: T[1,b,c,...,x,y,z] = 3
Recursive case: T[a,b,c,...,x,y,z] = T[T[a-1,b,c,...,x,y,z],b-1,c,...,x,y,z]
Reduction case: (array starts with 2) T[2,1,1,...,1,1,a,b,c,...,x,y,z] = T[3,3,3,...,3,3,a-1,b,c,...,x,y,z]
(Note: the number of ones can also be zero here, this works for any case that starts with 2)
Catastrophic case: T[2,1,...,1,1,z] = T[3,3,...,3,3,z-1]
Alt. representation:
T[a,b,c,...,x,y,z] = {3,a,b,c,...,x,y,z} in BEAF/BAN.