A302016 Expansion of 1/(1 - x - x^2/(1 + x^2/(1 + x^3/(1 + x^4/(1 + x^5/(1 + ...)))))), a continued fraction.

1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 46, 68, 102, 153, 229, 342, 510, 761, 1136, 1697, 2535, 3786, 5653, 8441, 12605, 18824, 28112, 41981, 62691, 93617, 139800, 208768, 311761, 465564, 695242, 1038226, 1550415, 2315284, 3457489, 5163181, 7710344, 11514102, 17194374, 25676907, 38344147

Offset: 0

Author

Ilya Gutkovskiy, Mar 30 2018

Keywords

• nonn

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Crossrefs

Antidiagonal sums of A291678.

Cf. A003823, A302015.

Programs

Mathematica

nmax = 44; CoefficientList[Series[1/(1 - x - x^2/(1 + ContinuedFractionK[x^k, 1, {k, 2, nmax}])), {x, 0, nmax}], x]
nmax = 44; CoefficientList[Series[1/(1 - x QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]/(QPochhammer[x, x^5] QPochhammer[x^4, x^5])), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 - x^(5*k-2))*(1 - x^(5*k-3))/((1 - x^(5*k-1))*(1 - x^(5*k-4)))).

a(0) = 1; a(n) = Sum_{k=1..n} A003823(k-1)*a(n-k).

a(n) ~ c / r^n, where r = 0.669643458685499460127124120930664114507093547265881... is the root of the equation x*QPochhammer[x^2, x^5]*QPochhammer[x^3, x^5] = QPochhammer[x, x^5]*QPochhammer[x^4, x^5] and c = 0.833333547701931811823757549354805979633827853516233646128015838266... - Vaclav Kotesovec, Jun 08 2019