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

0,3

Links

Table of n, a(n) for n=0..32.

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

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 *)

Prog

(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)

Crossrefs

Cf. A230823, A355043, A355046.

Cf. A088354.

Sequence in context: A370841 A063028 A085810 * A235611 A307789 A005773

Adjacent sequences: A355037 A355038 A355039 * A355041 A355042 A355043

Keyword

nonn

Author

Joerg Arndt, Jun 16 2022

Status

approved