User blog:Allam948736/Simplex function

I defined another extension of the idea of triangular, tetrahedral, pentatopic, ... numbers into the realm of googology, this time defining a function based off of these sequences, starting with:

S(1, n) = n

S(2, n) = trn(n) (or 1 + 2 + 3 + ... + n)

S(3, n) = trn(1) + trn(2) + trn(3) + ... + trn(n)

...

S(m, n) = S(m-1, 1) + S(m-1, 2) + S(m-1, 3) + ... + S(m-1, n)

To continue, I built an Ackermann-like function off of this, defined like so:

S(a, b, 1) = S(a, b)

S(a, b, c) = S(a, S(a, ...(a, S(a, b, c-1), c-1), ... c-1), c-1) w/ b nestings

This means that S(2, n, 2) is equal to trn^n(n), and S(3, n, 2) is equal to n "tetrahedroned" n times. While the triangular number googolisms I previously invented (triangrol, triangoogol, triangrolplex, great triangrol, and great triangrolplex) cannot be expressed concisely using this function without nesting the function, the triangrol can be expressed as S(2, S(2, S(2, 6, 2))).

s(2, 2, 2) = 6

s(2, 3, 2) = 231

s(2, 4, 2) = 1186570

s(2, 5, 2) = 347357071281165

s(2, 2, 3) = s(2, s(2, 2, 2), 2) s(2, 6, 2) = 2076895351339769460477611370186681 (2 triangled 8 times)

s(2, 3, 3) = s(2, s(2, s(2, 3, 2), 2), 2) = s(2, 231, 2) \(\approx 10^{10^{10^{7.1215\times10^{69}}\)

s(3, 2, 3) \(\approx 1.929016145\times10^{3252852494}\)

s(3, 3, 3) \(\approx 10^{10^{10^{10^{858312}}}}\)

s(4, 2, 2) = 70

s(4, 2, 3) = s(4, s(4, 2, 2), 2) = s(4, 70, 2) \(\approx 10^{1.943\times10^{42}}\)

s(5, 2, 2) = 252

s(5, 2, 3) = s(5, 252, 2)