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A378960 The "tetrational mean" of 2 and 3 determined as the mutual limit of interdependent sequences.

%I A378960 #22 Dec 30 2024 17:23:06
%S A378960 2,4,1,9,3,6,5,3,2,1,8,4,4,9,2,1,7,8,8,8,6,0,7,4,5,4,6,8,9,3,2,7,5,4,
%T A378960 3,5,4,4,1,6,4,6,2,6,2,4,3,6,8,7,9,3,9,1,4,5,5,7,2,2,8,4,7,0,1,1,2,0,
%U A378960 9,6,3,6,2,4,3,5,6,3,9,7,4,1,4,4,8,4,0,1,3,7,9,2,2,4,5,0,7,8,9,6,8,2,7,0,2,7,2,8,9,1,7,7,3,7,7
%N A378960 The "tetrational mean" of 2 and 3 determined as the mutual limit of interdependent sequences.
%C A378960 In an attempt to generalize the arithmetic mean (sum-based) and the geometric mean (product-based) to a similar construct for exponentiation, one can devise a simple definition using 2 interdependent sequences:
%C A378960   a_0 = x, b_0 = y,
%C A378960   a_n = exp(LambertW(log(a_{n-1}^b_{n-1}))),
%C A378960   b_n = exp(LambertW(log(b_{n-1}^a_{n-1}))), where x and y are the numbers for which we have to determine their "tetrational mean".
%C A378960 The averaging operation is the square super-root of each of the 2 possible exponentiation orders to give out the successive term of each defining sequence. The square super-root of x is exp(LambertW(log(x))) for a particular branch of the LambertW function.
%C A378960 If a_n and b_n converge to a number C then the "tetrational mean" of x and y is C. There may be a need to choose a particular branch of the LambertW function depending on the values of x and y (and that of log(x^y) and log(y^x)). This constant is based on the principal branch of the LambertW function.
%H A378960 Pham G. Hoang, <a href="/A378960/b378960.txt">Table of n, a(n) for n = 1..1000</a>
%H A378960 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_mean">Arithmetic mean</a>
%H A378960 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%H A378960 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mean">Mean</a>
%H A378960 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>
%e A378960 2.419365321844921788860745468932754354...
%Y A378960 Cf. A068985, A030798, A010464.
%K A378960 nonn,cons
%O A378960 1,1
%A A378960 _Pham G. Hoang_, Dec 12 2024