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A272853 Ramanujan's alpha-series.

%I A272853 #23 Jul 04 2023 12:15:40
%S A272853 9,791,65601,5444135,451797561,37493753471,3111529740489,
%T A272853 258219474707159,21429104870953665,1778357484814447079,
%U A272853 147582242134728153849,12247547739697622322431
%N A272853 Ramanujan's alpha-series.
%C A272853 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).
%D A272853 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272853 Seiichi Manyama, <a href="/A272853/b272853.txt">Table of n, a(n) for n = 0..520</a>
%H A272853 Robert Munafo, <a href="http://www.mrob.com/pub/seq/ramanujan.html">Sequences Related to the Work of Srinivasa Ramanujan</a>
%H A272853 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82, 82, -1).
%F A272853 G.f.: (9+53*x+x^2)/(1-82*x-82*x^2+x^3).
%F A272853 a(-3)=-11161; a(-2)=-135; a(-1)=-1; a(n) = 82*a(n-1)+82*a(n-2)-a(n-3).
%F A272853 A272853(n)^3 + A272854(n)^3 = A272855(n)^3 + (-1)^n.
%e A272853 a(3)=5444135 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%t A272853 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 *)
%o A272853 (Wolfram|Alpha) Series[(1+53*a+9*a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272853 Cf. A051028, A051029, A051030.
%Y A272853 Cf. A272854, A272855.
%K A272853 nonn
%O A272853 0,1
%A A272853 _Robert Munafo_, May 08 2016