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A280918 2nd term of the continued fraction for 2-sqrt(2)^^n, where x^^n denotes tetration. 1

%I #8 Oct 11 2018 17:35:14

%S 1,2,4,6,9,13,20,29,42,61,88,128,184,267,385,556,803,1159,1672,2413,

%T 3481,5023,7247,10456,15085,21764,31399,45299,65354,94286,136026,

%U 196245,283122,408459,589282,850155,1226515,1769487,2552830,3682956,5313383,7665592

%N 2nd term of the continued fraction for 2-sqrt(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 lim_{n->inf} sqrt(2)^^n = 2. This sequence shows the speed of convergence to this limit.

%H G. C. Greubel, <a href="/A280918/b280918.txt">Table of n, a(n) for n = 1..300</a>

%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(n) ~ 1/(A277435*log(2)^n).

%t Table[ContinuedFraction[2 - Power@@Table[Sqrt[2], {n}], 2][[2]], {n, 42}]

%Y Cf. A198094, A277435.

%K nonn

%O 1,2

%A _Vladimir Reshetnikov_, Jan 10 2017

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Last modified March 1 06:29 EST 2024. Contains 370430 sequences. (Running on oeis4.)