A051028 Ramanujan's a-series: expansion of (1+53x+9x^2)/(1-82x-82x^2+x^3).

1, 135, 11161, 926271, 76869289, 6379224759, 529398785665, 43933719985479, 3645969360009049, 302571523160765631, 25109790452983538281, 2083810036074472911735, 172931123203728268135681, 14351199415873371782349831, 1190976620394286129666900249

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

The "amazing" identity of Ramanujan is a(n)^3 + b(n)^3 = c(n)^3 + (-1)^n, where a(n)=A051028(n), b(n)=A051029(n) and c(n)=A051030(n). - Emeric Deutsch, Oct 14 2006

Links

• Robert Israel, Table of n, a(n) for n = 0..468
• Kwang-Wu Chen, Extensions of an amazing identity of Ramanujan, Fib. Q., 50 (2012), 227-230.
• Jung Hun Han and Michael D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304.
• Michael D. Hirschhorn, An amazing identity of Ramanujan, Math. Mag. 68 (1995), no. 3, 199--201. MR1335148
• Michael D. Hirschhorn, A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan, Math. Mag., 69.4 (1996), 267-269.
• Michael D. Hirschhorn, Ramanujan and Fermat's Last Theorem, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
• J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
• Eric Rowland and Jesus Sistos Barron, Complexity of powers of a constant-recursive sequence, arXiv:2501.14643 [math.NT], 2025. See p. 2.
• Eric Weisstein's World of Mathematics, Ramanujan's Sum Identity.
• Index entries for linear recurrences with constant coefficients, signature (82,82,-1).

Crossrefs

Cf. A051029, A051030.

Programs

Maple

g:=(1+53*x+9*x^2)/(1-82*x-82*x^2+x^3): gser:=series(g,x=0,20): seq(coeff(gser,x,n),n=0..12); # Emeric Deutsch, Oct 14 2006

Mathematica

CoefficientList[Series[(1 + 53 x + 9 x^2)/(1 - 82 x - 82 x^2 + x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 22 2015 *)

PARI

Vec((1+53*x+9*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016

Formula

G.f.: (1+53*x+9*x^2)/((1+x)*(1-83*x+x^2)).

X(n+1) = A*X(n), where X(n) = transpose(A051028(n), A051029(n), A051030(n)) and A = matrix(3,3,[63,104,-68; 64,104,-67; 80,131,-85]). - Emeric Deutsch, Oct 14 2006

Extensions

Minor edits (g.f. and name) by M. F. Hasler, May 08 2016