A272853 Ramanujan's alpha-series.

9, 791, 65601, 5444135, 451797561, 37493753471, 3111529740489, 258219474707159, 21429104870953665, 1778357484814447079, 147582242134728153849, 12247547739697622322431

Offset

0,1

Comments

Ramanujan's notes define this by the same G.f. as A051028 (the a-series) but using Laurent series expansion. 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).

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S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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).

Formula

G.f.: (9+53*x+x^2)/(1-82*x-82*x^2+x^3).

a(-3)=-11161; a(-2)=-135; a(-1)=-1; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).

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

Example

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

Mathematica

Rest@ CoefficientList[Normal@ Series[(1 + 53*a + 9*a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 20}], 1/a] (* Giovanni Resta, May 08 2016 *)

Prog

(Wolfram|Alpha) Series[(1+53*a+9*a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]

Crossrefs

Cf. A051028, A051029, A051030.

Cf. A272854, A272855.

Sequence in context: A168257 A303143 A137065 * A332179 A196981 A197166

Adjacent sequences: A272850 A272851 A272852 * A272854 A272855 A272856

Keyword

nonn

Author

Robert Munafo, May 08 2016

Status

approved