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

1, 2, 4, 6, 9, 13, 20, 29, 42, 61, 88, 128, 184, 267, 385, 556, 803, 1159, 1672, 2413, 3481, 5023, 7247, 10456, 15085, 21764, 31399, 45299, 65354, 94286, 136026, 196245, 283122, 408459, 589282, 850155, 1226515, 1769487, 2552830, 3682956, 5313383, 7665592

Offset: 1

Author

Vladimir Reshetnikov, Jan 10 2017

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..300
• Eric Weisstein's World of Mathematics, Power Tower
• Wikipedia, Tetration

Crossrefs

Cf. A198094, A277435.

Programs

Mathematica

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

Formula

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