A355040 Expansion of the continued fraction 1 / (1-q / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...))))).
1, 1, 2, 5, 13, 35, 96, 266, 742, 2079, 5843, 16457, 46423, 131099, 370527, 1047858, 2964698, 8390837, 23754234, 67260645, 190478213, 539484321, 1528094423, 4328632609, 12262352881, 34738763766, 98416624789, 278825903115, 789961599608, 2238129694407, 6341171821627, 17966261019890, 50903653156245
Offset: 0
Keywords
Programs
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Mathematica
nmax = 40; CoefficientList[Series[1/(1 - x/(1 - x + ContinuedFractionK[-x^k, 1 - x*(1 - x^k)/(1 - x), {k, 2, nmax}])), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2022 *) A355040List[nmax_] := Module[{a, b, q}, a = ContinuedFractionK[-q^k, (q^(k + 1) - 2 q + 1)/(q - 1), {k, 0, nmax}]; b = Series[a, {q, 0, nmax}]; CoefficientList[b, q] ]; A355040List[32] (* Peter Luschny, Jun 20 2022 *) -
PARI
N=44; q='q+O('q^N); f(n) = 1 - sum(k=1,n-1,q^k); s=1; forstep(j=N, 1, -1, s = q^j/s; s = f(j) - s ); s = 1/s; Vec(s)
Formula
a(n) ~ c * d^n, where d = 2.8333645039948803621658813720055524872119... and c = 0.1710130167563241590871776261530008679... - Vaclav Kotesovec, Jun 16 2022
a(n) = [q^n] K_{k>=0} -q^k / ((q^(k + 1) - 2*q + 1)/(q - 1)), where K is Gauss's notation for continued fractions. - Peter Luschny, Jun 20 2022