A277435 - OEIS

%I #24 Oct 19 2016 08:01:54

%S 6,3,2,0,9,8,6,6,1,0,5,0,8,2,9,2,5,0,3,5,5,4,5,0,6,4,5,9,9,0,7,8,0,8,

%T 6,2,7,9,9,4,7,4,5,5,2,3,2,4,1,6,4,4,7,9,6,6,9,7,2,3,3,8,2,4,3,2,5,8,

%U 6,1,6,2,7,6,1,5,0,9,6,2,1,4,7,0,9,1,7,6,6,4,9,4,2,6,6,7,7,3,7,1,6,3,7,9,4,6

%N Decimal expansion of lim_{n->inf} (2 - sqrt(2)^^n)/log(2)^n, where x^^n denotes tetration.

%C Tetration x^^n is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Note that sqrt(2)^^inf = lim_{n->inf} sqrt(2)^^n = 2. Asymptotically, sqrt(2)^^n = 2 - O(log(2)^n). This constant is the coefficient in the O(log(2)^n) term. Furthermore, sqrt(2)^^n = 2 - a*log(2)^n + (a^2/(4*(1 - 1/log(2))))*log(2)^(2*n) + O(log(2)^(3*n)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetration">Tetration</a>

%F a = 2*sqrt(2)*A260691/(1-log(2)).

%e 0.63209866105082925035545064599078...

%t RealDigits[SequenceLimit[1`200 Table[(2 - Power @@ Table[Sqrt[2], {n}])/Log[2]^n, {n, 1, 200}]], 10, 100][[1]]

%Y Cf. A198094, A260691.

%K nonn,cons

%O 0,1

%A _Vladimir Reshetnikov_, Oct 14 2016