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.
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 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) = 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
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
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Minor edits (g.f. and name) by M. F. Hasler, May 08 2016
STATUS
approved