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A272854 Ramanujan's beta-series. 3
10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018, 265305806511788, 22017185020849402, 1827161050923988562, 151632350041670201260, 12583657892407702716002 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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

EXAMPLE

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

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 *)

PROG

(Wolfram|Alpha) Series[-1*(2+8a-10a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]

CROSSREFS

Cf. A051028, A051029, A051030.

Cf. A272853, A272855.

Sequence in context: A347845 A006440 A054944 * A015093 A104906 A013386

Adjacent sequences: A272851 A272852 A272853 * A272855 A272856 A272857

KEYWORD

nonn,easy

AUTHOR

Robert Munafo, May 08 2016

STATUS

approved

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Last modified December 5 18:51 EST 2022. Contains 358588 sequences. (Running on oeis4.)