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A260691 Decimal expansion of a constant related to asymptotic behavior of super-roots of 2: lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.

%I A260691 #17 Feb 16 2025 08:33:26
%S A260691 0,6,8,5,7,5,6,5,9,8,1,1,3,2,9,1,0,3,9,7,6,5,5,3,3,1,1,4,1,5,5,0,6,5,
%T A260691 5,4,2,3,3,5,6,3,5,7,1,3,7,8,6,1,9,4,4,7,4,6,8,1,2,5,1,7,0,5,1,0,3,4,
%U A260691 8,4,4,6,8,0,7,3,4,9,7,3,7,7,4,6,0,7,1,7,1,4,3,0,9,3,0,8,1,9,7,9,1,1,1,3,9,7,4,2,8,6
%N A260691 Decimal expansion of a constant related to asymptotic behavior of super-roots of 2: lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.
%C A260691 Tetration is defined recursively: x^^0 = 1, x^^n = x^(x^^(n-1)). Its inverse, super-root, is defined: sr[n](y) = x iff x^^n = y. Note that lim_{n->inf} sr[n](2) = sqrt(2). Asymptotically, sr[n](2) = sqrt(2) + O(log(2)^n). This constant is the coefficient in the O(log(2)^n) term, i.e. lim_{n->inf} (sr[n](2) - sqrt(2))/log(2)^n.
%H A260691 G. C. Greubel, <a href="/A260691/b260691.txt">Table of n, a(n) for n = 0..193</a>
%H A260691 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%H A260691 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetration#Super-root">Super-root</a>
%F A260691 a = A277435*(1-log(2))/(2*sqrt(2)). - _Vladimir Reshetnikov_, Oct 18 2016
%e A260691 0.0685756598113291039765533114155...
%t A260691 {0}~Join~RealDigits[SequenceLimit[1`200 Table[(2 - Power @@ Table[Sqrt[2], {n}])/Log[2]^n, {n, 1, 200}]] (1 - Log[2])/(2 Sqrt[2]), 10, 100][[1]] (* _Vladimir Reshetnikov_, Oct 18 2016 *)
%Y A260691 Cf. A198094, A277435.
%K A260691 cons,nonn
%O A260691 0,2
%A A260691 _Vladimir Reshetnikov_, Nov 15 2015