|
%I #47 Jun 17 2023 22:55:33
%S 1,135,11161,926271,76869289,6379224759,529398785665,43933719985479,
%T 3645969360009049,302571523160765631,25109790452983538281,
%U 2083810036074472911735,172931123203728268135681,14351199415873371782349831,1190976620394286129666900249
%N Ramanujan's a-series: expansion of (1+53x+9x^2)/(1-82x-82x^2+x^3).
%C 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
%H Robert Israel, <a href="/A051028/b051028.txt">Table of n, a(n) for n = 0..468</a>
%H Kwang-Wu Chen, <a href="https://www.fq.math.ca/Papers1/50-3/Kwang-WuChen.pdf">Extensions of an amazing identity of Ramanujan</a>, Fib. Q., 50 (2012), 227-230.
%H J. H. Han and M. D. Hirschhorn, <a href="http://www.jstor.org/stable/27642956">Another Look at an Amazing Identity of Ramanujan</a>, Mathematics Magazine, Vol. 79 (2006), pp. 302-304.
%H Michael D. Hirschhorn, <a href="http://www.jstor.org/stable/2691416">An amazing identity of Ramanujan</a>, Math. Mag. 68 (1995), no. 3, 199--201. MR1335148
%H Michael D. Hirschhorn, <a href="http://www.jstor.org/stable/2690530">A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan</a>, Math. Mag., 69.4 (1996), 267-269.
%H M. D. Hirschhorn, <a href="http://www.austms.org.au/wp-content/uploads/Gazette/2004/Sep04/hirschhorn.pdf">Ramanujan and Fermat's Last Theorem</a>, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
%H J. Mc Laughlin, <a href="http://www.fq.math.ca/Papers1/48-1/McLaughlin.pdf">An identity motivated by an amazing identity of Ramanujan</a>, Fib. Q., 48 (No. 1, 2010), 34-38.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujansSumIdentity.html">Ramanujan's Sum Identity.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82,82,-1).
%F G.f.: (1+53*x+9*x^2)/((1+x)*(1-83*x+x^2)).
%F 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
%p 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
%t 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 *)
%o (PARI) Vec((1+53*x+9*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ _Michel Marcus_, Feb 29 2016
%Y Cf. A051029, A051030.
%K nonn,easy
%O 0,2
%A _Eric W. Weisstein_
%E Minor edits (g.f. and name) by _M. F. Hasler_, May 08 2016
|
