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

0,3

Links

Table of n, a(n) for n=0..44.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

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

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]

Crossrefs

Antidiagonal sums of A291678.

Cf. A003823, A302015.

Sequence in context: A039884 A240496 A212464 * A078620 A073941 A005428

Adjacent sequences: A302013 A302014 A302015 * A302017 A302018 A302019

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 30 2018

Status

approved