A355043 Expansion of the continued fraction 1 / (1-q-q^2 / (1-q-q^2-q^3 / (1-q-q^2-q^3-q^4 / (...)))).

1, 1, 2, 4, 9, 21, 50, 121, 296, 730, 1811, 4513, 11285, 28294, 71088, 178904, 450840, 1137345, 2871720, 7256093, 18345060, 46403039, 117421762, 297232446, 752601692, 1906056161, 4828267801, 12232594912, 30996034963, 78549710061, 199079279640, 504596195477, 1279065489044

Offset: 0

Author

Joerg Arndt, Jun 16 2022

Keywords

• nonn

Comments

Starts similar to A091964, terms differ after 730.

Crossrefs

Cf. A355040, A355046, A091964.

Programs

Mathematica

nmax = 40; CoefficientList[Series[1/(1 - x - x^2/(1 - x - x^2 + ContinuedFractionK[-x^k, 1 - x*(1 - x^k)/(1 - x), {k, 3, nmax}])), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2022 *)

PARI

N=44; q='q+O('q^N);
f(n) = 1 - sum(k=1,n-1,q^k);
s=1; forstep(j=N, 2, -1, s = q^j/s; s = f(j) - s ); s = 1/s;
Vec(s)

Formula

a(n) ~ c * d^n, where d = 2.5358790673564851880281667369326354455... and c = 0.14917782209027525483339419811881753... - Vaclav Kotesovec, Jun 16 2022