A308745 Expansion of 1/(1 - x*(1 + x)/(1 - x^2*(1 + x^2)/(1 - x^3*(1 + x^3)/(1 - x^4*(1 + x^4)/(1 - ...))))), a continued fraction.

1, 1, 2, 4, 8, 17, 36, 76, 161, 342, 726, 1542, 3276, 6960, 14788, 31422, 66767, 141872, 301464, 640584, 1361188, 2892417, 6146164, 13060136, 27751818, 58970564, 125308114, 266270558, 565805452, 1202295228, 2554789536, 5428741218, 11535678790, 24512475453

Offset

0,3

Links

Table of n, a(n) for n=0..33.

Formula

From Vaclav Kotesovec, Jun 25 2019: (Start)

a(n) ~ c * d^n, where

d = 2.124927028900893046638236231387101475346473032396641627320401...

c = 0.386397654364351443933577245182777062935616240164642598839093... (End)

From Peter Bala, Dec 18 2020

Conjectural g.f.: 1/(2 - (1 + x)/(1 - x^2/(2 - (1 + x^3)/(1 - x^4/(2 - (1 + x^5)/(1 - x^6/(2 - ... ))))))).

More generally it appears that 1/(1 - t*x*(1 + u*x)/(1 - t*x^2*(1 + u*x^2)/(1 - t*x^3*(1 + u*x^3)/(1 - t*x^4*(1 + u*x^4)/(1 - ... ))))) = 1/(1 + u - (u + t*x)/(1 - t*x^2/(1 + u - (u + t*x^3)/(1 - t*x^4/(1 + u - (u + t*x^5)/(1 - ... )))))). (End)

Mathematica

nmax = 33; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^k (1 + x^k), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A005169, A053254, A092848, A143064.

Sequence in context: A292322 A008999 A052903 * A367714 A226729 A063457

Adjacent sequences: A308742 A308743 A308744 * A308746 A308747 A308748

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jun 21 2019

Status

approved