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A051028 Ramanujan's a-series: expansion of (1+53x+9x^2)/(1-82x-82x^2+x^3). 7
1, 135, 11161, 926271, 76869289, 6379224759, 529398785665, 43933719985479, 3645969360009049, 302571523160765631, 25109790452983538281, 2083810036074472911735, 172931123203728268135681, 14351199415873371782349831, 1190976620394286129666900249 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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.

M. D. Hirschhorn, Ramanujan and Fermat's Last Theorem, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.

J. McLaughlin, 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.: (1+53*x+9*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:=(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 *)

PROG

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

CROSSREFS

Cf. A051029, A051030.

Sequence in context: A004005 A248009 A143404 * A076011 A132054 A273440

Adjacent sequences: A051025 A051026 A051027 * A051029 A051030 A051031

KEYWORD

nonn,easy

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

STATUS

approved

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Last modified December 5 19:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)