A306576 Expansion of 1/(1 - x - 2*x/(1 - 2*x - 3*x/(1 - 3*x - 4*x/(1 - 4*x - 5*x/(1 - ...))))), a continued fraction.

1, 3, 19, 179, 2183, 32355, 562343, 11198203, 251297263, 6275390067, 172639089031, 5189033793611, 169220733646271, 5951777459480931, 224604052936701815, 9053124776482735291, 388198017158108201839, 17645733672934447166163, 847577245047341210277415

Offset: 0

Author

Ilya Gutkovskiy, Jun 25 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A000311, A001147, A003319, A158882.

Programs

Mathematica

$RecursionLimit = Infinity; nmax = 18; CoefficientList[Series[1/(1 - x + ContinuedFractionK[-k x, 1 - k x, {k, 2, nmax + 1}]), {x, 0, nmax}], x]

Formula

a(n) ~ c * d^n * n^(n+1), where d = 1 / (exp(1) * (2*log(2) - 1)) = 0.952329306865721945... and c = 1/(sqrt(2) * (2*log(2) - 1)^(3/2)) = 2.945150206105358... - Vaclav Kotesovec, Jul 01 2019, updated May 06 2024