A272854 Ramanujan's beta-series.

10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018, 265305806511788, 22017185020849402, 1827161050923988562, 151632350041670201260, 12583657892407702716002

Offset: 0

Author

Robert Munafo, May 08 2016

Keywords

• nonn
• easy

Comments

Ramanujan's notes define this by the same G.f. as A051030 (the c-series) but using Laurent series expansion. It is mislabeled as "gamma" in Ramanujan's notes. These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n). They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).

Examples

			a(3)=5593538 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
		

References

• S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..520
• Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
• Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Crossrefs

Cf. A051028, A051029, A051030.

Cf. A272853, A272855.

Programs

Mathematica

Rest@ CoefficientList[ Normal@Series[-(2 + 8*x - 10*x^2)/(1 - 82*x - 82*x^2 + x^3), {x, Infinity, 20}], 1/x] (* Giovanni Resta, May 08 2016 *)

Formula

G.f.: (10-8*x-2*x^2)/(1-82*x-82*x^2+x^3).

a(-3)=14258; a(-2)=172; a(-1)=2; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).

A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.