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 REFERENCES S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341.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

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Last modified October 4 20:53 EDT 2023. Contains 365888 sequences. (Running on oeis4.)