A051029 Ramanujan's b-series: expansion of (2-26x-12x^2)/(1-82x-82x^2+x^3).

2, 138, 11468, 951690, 78978818, 6554290188, 543927106802, 45139395574362, 3746025905565260, 310875010766342202, 25798879867700837522, 2140996154008403172108, 177676881902829762447458, 14745040201780861879966890, 1223660659865908706274804428

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

References

For additional references and links see A051028.

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.

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.: (2-26*x-12*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

Maple

g:=(2-26*x-12*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 - 26 x - 12 x^2)/(1 - 82 x - 82 x^2 + x^3), {x, 0, 33}], x] (* Vincenzo Librandi, Jul 22 2015 *)

Prog

(PARI) Vec((2-26*x-12*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ Michel Marcus, Feb 29 2016

Crossrefs

Cf. A051028, A051030.

Sequence in context: A221227 A087619 A157072 * A212715 A329594 A084560

Adjacent sequences: A051026 A051027 A051028 * A051030 A051031 A051032

Keyword

nonn

Author

Eric W. Weisstein

Extensions

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

Status

approved