A355040 -id:A355040 - cp's OEIS frontend

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

Joerg Arndt, Jun 16 2022

nonn

Starts similar to A091964, terms differ after 730.

Cf. A355040, A355046, A091964.

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 *)

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)

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

A355046 Expansion of the continued fraction 1 / (1-q / (1-q-q^3 / (1-q-q^3-q^5 / (1-q-q^3-q^5-q^7 / (...))))).

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 49, 115, 271, 640, 1514, 3585, 8494, 20134, 47740, 113221, 268557, 637077, 1511402, 3585843, 8507837, 20186405, 47896899, 113648058, 269662860, 639857869, 1518267397, 3602589217, 8548353709, 20283885193, 48130511518, 114206363723, 270994509775, 643029572029

Offset: 0

Joerg Arndt, Jun 16 2022

nonn

Cf. A355040, A355043.

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

cp's OEIS Frontend

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 / (...)))).

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