%I #62 Jan 11 2020 15:57:46

%S 3,4,8,2048

%N a(n) = w_n(2), Wainer's fast-growing hierarchy function applied to two.

%C From _Natan Arie Consigli_, Oct 09 2016: (Start)

%C Wainer's fast-growing function is defined as follows:

%C w_0(x) = x+1;

%C w_n+1(x) = (w_n)^x(x) = w_n(w_n(...w_n(x)) (x iterations);

%C This function is a particular instance of the fast-iteration hierarchy function F[k]_n(x). Wainer's fast-growing function is F[1]_n(x). See A275000 for details and definitions.

%C Because of its simple definition, this function is a popular benchmark for large number functions. (End)

%H Googology Wiki, <a href="http://googology.wikia.com/wiki/Fast-growing_hierarchy">Fast Growing Hierarchy</a>

%H Project Euclid, <a href="http://projecteuclid.org/euclid.lnl/1235415914">Wainer:Accessible Segments of the Fast Growing Hierarchy</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fast-growing_hierarchy">Fast-growing hierarchy</a>

%F w_0(x) = x+1;

%F w_1(x) = (((x+1)+1)+...+1) = x+x = 2*x;

%F w_2(x) = 2(2(...(2(2x))) = 2^x*x;

%F w_3(x) = Product_{k=0...x} b(k), where b(0)=x, b(m+1) = 2^Product_{l=0...m} b(l). - _Benoit Jubin_, Jan 15 2009; edited by _Natan Arie Consigli_, Oct 09 2016

%e Using formula:

%e a(0) = w_0(2) = 1+2 = 3;

%e a(1) = w_1(2) = 2*2 = 4;

%e a(2) = w_2(2) = 2^2*2 = 8;

%e a(3) = w_3(2) = Product_{k=0...2} b(k) = b(0)*b(1)*b(2) = 2 * 2^b(0) * 2^(b(1)*b(0)) = 2 * 2^2 * 2^(2^2*2) = 2048;

%e Using recursion:

%e a(0) = w_0(2) = 1+2 = 3;

%e a(1) = w_1(2) = w_0(w_0(2)) = w_0(3) = 3+1 = 4;

%e a(2) = w_2(2) = w_1(w_1(2)) = w_1(4) = (w_0)^4(4) = 4+1+1+1+1 = 8;

%e a(3) = w_3(2) = w_2(w_2(2)) = w_2(8) = (w_1)^8(8) = ... = 2048;

%e a(4) is obtained by applying the operation w_2(x) 2048 times to 2048, where w_2(2^p) = 2^( 2^p+p ). Thus a(5) is larger than 2^(2^(...2^(2^11+11)...)), with 2049(?) occurrences of "2^". - _M. F. Hasler_, Jan 15 2009; edited by _Natan Arie Consigli_, Oct 09 2016

%t w[0, x_] := x + 1; w[n_, x_] := Nest[w[n - 1, # ]&, x, x]; Table[w[n, 2], {n, 0, 3}] (* edited by _Natan Arie Consigli_, Oct 09 2016 *)

%Y Cf. A275000 (fast-iteration function applied to two).

%K nonn,nice,bref

%O 0,1

%A _Vladimir Reshetnikov_, Jan 14 2009

%E Revised by _Natan Arie Consigli_, Apr 03 2016, Oct 09 2016