A051030 Ramanujan's c-series: expansion of (2+8*x-10*x^2)/(1-82*x-82*x^2+x^3).

2, 172, 14258, 1183258, 98196140, 8149096378, 676276803218, 56122825570732, 4657518245567522, 386517891556533610, 32076327480946722092, 2661948663027021400042, 220909662703761829481378, 18332840055749204825554348, 1521404814964480238691529490

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

• Harvey P. Dale, Table of n, a(n) for n = 0..500
• 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.
• 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. A051028, A051029.

Programs

Magma

I:=[2,172,14258]; [n le 3 select I[n] else 82*Self(n-1)+82*Self(n-2)-Self(n-3):n in [1..30]]; // Vincenzo Librandi, Feb 29 2016

Maple

g:=(2+8*x-10*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[(2+8x-10x^2)/(1-82x-82x^2+x^3),{x,0,30}],x] (* or *) LinearRecurrence[{82,82,-1},{2,172,14258},20] (* Harvey P. Dale, Dec 17 2012 *)

PARI

Vec((2+8*x-10*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016

Formula

G.f.: (2+8*x-10*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

a(0) = 2, a(1) = 172, a(2) = 14258, a(n) = 82*a(n-1)+82*a(n-2)-a(n-3). - Harvey P. Dale, Dec 17 2012

Extensions

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