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A275000 a(n) = F[n]_n(2), main diagonal of fast-iteration function applied to 2. 9
2, 4, 18, 590295810358705651712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The next term is too large to include.

The fast-iteration (or extended Grzegorczyk hierarchy) function F[k]_n(x) is defined as follows:

F[k]_{n+1}(x) = (F[k]_n)^x(x) = F[k]_n(F[k]_n(...F[k]_n(x)) (with x iterations);

F[k]_0(x) = x+k.

The base case could be rewritten using n=1 rather than n=0. If so the definition would be:

F'[k]_n+1(x) = (F'[k]_n)^x(x);

F'[k]_1(x) = x+k.

Because of its clear definition, this function is a popular benchmark for large number functions.

LINKS

Table of n, a(n) for n=0..3.

Googology Wiki, Fast Growing Hierarchy

Wikipedia, Fast-growing hierarchy.

FORMULA

For small values of n we have:

F[k]_0(x) = x+k;

F[k]_1(x) = x+kx = (k+1)x;

F[k]_2(x) = x(k+1)^x.

EXAMPLE

F[0]_0(2) = 2+0 = 2;

F[1]_1(2) = (1+1)2 = 4;

F[2]_2(2) = 2(2+1)^2 = 18;

F[3]_3(2) = F[3]_2(F[3]_2(2)) = F[3]_2(2(3+1)^2) = F[3]_2(32) = 32(3+1)^32 = 590295810358705651712.

MATHEMATICA

f[k_, 0, x_] := x + k; f[k_, n_, x_] := Nest[f[k, n - 1, # ]&, x, x]; Table[f[n, n, 2], {n, 0, 3}]

CROSSREFS

A154714 gives w_n(2) = F[1]_n(2).

Sequence in context: A318531 A009667 A009508 * A290595 A009418 A273553

Adjacent sequences:  A274997 A274998 A274999 * A275001 A275002 A275003

KEYWORD

nonn

AUTHOR

Natan Arie Consigli, Oct 08 2016

STATUS

approved

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Last modified December 2 23:22 EST 2021. Contains 349445 sequences. (Running on oeis4.)