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A272853 Ramanujan's alpha-series. 3
9, 791, 65601, 5444135, 451797561, 37493753471, 3111529740489, 258219474707159, 21429104870953665, 1778357484814447079, 147582242134728153849, 12247547739697622322431 (list; graph; refs; listen; history; text; internal format)
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S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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
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
Sequence in context: A168257 A303143 A137065 * A332179 A196981 A197166
KEYWORD
nonn
AUTHOR
Robert Munafo, May 08 2016
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)