login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A285637 G.f.: 1/( (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. 3
1, 0, 1, 2, 1, 4, 6, 10, 19, 30, 55, 92, 161, 282, 483, 846, 1462, 2538, 4409, 7642, 13276, 23032, 39977, 69394, 120426, 209036, 362800, 629698, 1092952, 1896968, 3292522, 5714678, 9918752, 17215620, 29880461, 51862438, 90015657, 156236814, 271174435, 470667300, 816919764, 1417897172, 2460991365 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..42.

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

FORMULA

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

a(n) ~ c * d^n, where d = 1.7356628245303474256582607497196685302546528472903927546099... and c = 0.215558365582078354136603033062960103377669... (for the constant "d" see A168445). - Vaclav Kotesovec, Aug 26 2017

EXAMPLE

G.f.: A(x) = 1 + x^2 + 2*x^3 + x^4 + 4*x^5 + 6*x^6 + 10*x^7 + 19*x^8 + 30*x^9 + 55*x^10 + ...

MATHEMATICA

nmax = 42; 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 = 42; CoefficientList[Series[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 (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, 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, A285636, A285638.

Sequence in context: A048213 A283309 A054408 * A205845 A034424 A095012

Adjacent sequences:  A285634 A285635 A285636 * A285638 A285639 A285640

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Apr 23 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 9 05:17 EDT 2020. Contains 335538 sequences. (Running on oeis4.)