A285636 G.f.: (1 + x/(1 + x^2/(1 + x^3/(1 + x^4/(1 + ...))))) / (1 - x/(1 - x^2/(1 - x^3/(1 - x^4/(1 - ...))))), a continued fraction.

1, 2, 2, 2, 4, 8, 14, 22, 36, 64, 114, 198, 340, 586, 1018, 1772, 3076, 5332, 9248, 16054, 27872, 48376, 83952, 145700, 252888, 438938, 761846, 1322286, 2295022, 3983384, 6913822, 12000054, 20828006, 36150354, 62744812, 108903838, 189020310, 328075444, 569428264, 988335418, 1715417004

Offset

0,2

References

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 381.

Links

Robert Israel, Table of n, a(n) for n = 0..600

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

Formula

G.f.: A(x) = P(x)/(R(x)*Q(x)), where P(x) = Sum_{k>=0} (-1)^k*x^(k*(k+1)) / Product_{m=1..k} (1 - x^m), R(x) = Product_{k>=1} (1 - x^(5*k-1))*(1 - x^(5*k-4)) / ((1 - x^(5*k-2))*(1 - x^(5*k-3)) and Q(x) = Sum_{k>=0} (-1)^k*x^(k^2) / Product_{m=1..k} (1 - x^m).

a(n) ~ c / r^n, where r = A347901 = 0.576148769142756602297868573719938782354724663118974... is the lowest root of the equation Sum_{k>=0} (-1)^k * r^(k^2) / QPochhammer(r, r, k) = 0 and c = 0.452642356466453742995961374156022446123012... - Vaclav Kotesovec, Aug 26 2017, updated Sep 24 2020

Example

G.f.: A(x) = 1 + 2*x + 2*x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 14*x^6 + 22*x^7 + 36*x^8 + 64*x^9 + ...

Maple

A10:= [1, seq([x^i, 1], i=1..10)]: B10:= [1, seq([-x^i, 1], i=1..10)]:
S:= series(numtheory:-nthconver(A10, 10)/numtheory:-nthconver(B10, 10), x, 51):
A:= [seq(coeff(S, x, i), i=0..50)]; # Robert Israel, Dec 15 2024

Mathematica

nmax = 40; CoefficientList[Series[(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}]  Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}]), {x, 0, nmax}], x]

Crossrefs

Cf. A003823, A005169, A007325, A055101, A285635, A285637, A285638.

Sequence in context: A302613 A295680 A099768 * A102831 A262568 A183388

Adjacent sequences: A285633 A285634 A285635 * A285637 A285638 A285639

Keyword

nonn,changed

Author

Ilya Gutkovskiy, Apr 23 2017

Status

approved