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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.
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%I #12 Feb 03 2021 23:22:54

%S 1,1,2,4,8,17,36,76,161,342,726,1542,3276,6960,14788,31422,66767,

%T 141872,301464,640584,1361188,2892417,6146164,13060136,27751818,

%U 58970564,125308114,266270558,565805452,1202295228,2554789536,5428741218,11535678790,24512475453

%N 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.

%F From _Vaclav Kotesovec_, Jun 25 2019: (Start)

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

%F d = 2.124927028900893046638236231387101475346473032396641627320401...

%F c = 0.386397654364351443933577245182777062935616240164642598839093... (End)

%F From _Peter Bala_, Dec 18 2020

%F 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 - ... ))))))).

%F 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)

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

%Y Cf. A005169, A053254, A092848, A143064.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Jun 21 2019