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

0,1

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.

J. H. Han and M. 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 Weisstein's World of Mathematics, Ramanujan's Sum Identity.

Index entries for linear recurrences with constant coefficients, signature (82, 82, -1).

Formula

G.f.: (2+8*x-10*x^2)/((1+x)*(1-83*x+x^2)).

X(n+1) = AX(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

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 *)

Prog

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

Crossrefs

Cf. A051028, A051029.

Sequence in context: A230511 A209607 A243230 * A339641 A262728 A139935

Adjacent sequences: A051027 A051028 A051029 * A051031 A051032 A051033

Keyword

nonn

Author

Eric W. Weisstein

Extensions

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

Status

approved