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A272855 Ramanujan's gamma-series. 3
12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140, 329860675188578, 27374461238587818, 2271750422127600332, 188527910575352239722, 15645544827332108296610 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Ramanujan's notes define this by the same G.f. as A051029 (the b-series) but using Laurent series expansion. It is mislabeled as "beta" 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).

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S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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

FORMULA

G.f.: x*(12+26*x-2*x^2)/(1-82*x-82*x^2+x^3).

a(-3)=11468; a(-2)=138; 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.

EXAMPLE

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

MATHEMATICA

Rest@ CoefficientList[ Normal@ Series[-1*(2 - 26 a - 12 a^2)/(1 - 82*a - 82*a^2 + a^3), {a, Infinity, 10}], 1/a] (* Giovanni Resta, May 08 2016 *)

PROG

(Wolfram|Alpha) Series[-1*(2-26a-12a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]

CROSSREFS

Cf. A051028, A051029, A051030.

Cf. A272853, A272854.

Sequence in context: A079916 A204327 A123283 * A265488 A232912 A004812

Adjacent sequences:  A272852 A272853 A272854 * A272856 A272857 A272858

KEYWORD

nonn

AUTHOR

Robert Munafo, May 08 2016

STATUS

approved

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Last modified April 5 08:04 EDT 2020. Contains 333238 sequences. (Running on oeis4.)