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Ramanujan's c-series: expansion of (2+8*x-10*x^2)/(1-82*x-82*x^2+x^3).
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%I #41 Jun 17 2023 22:55:42

%S 2,172,14258,1183258,98196140,8149096378,676276803218,56122825570732,

%T 4657518245567522,386517891556533610,32076327480946722092,

%U 2661948663027021400042,220909662703761829481378,18332840055749204825554348,1521404814964480238691529490

%N Ramanujan's c-series: expansion of (2+8*x-10*x^2)/(1-82*x-82*x^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 Harvey P. Dale, <a href="/A051030/b051030.txt">Table of n, a(n) for n = 0..500</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 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.: (2+8*x-10*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

%F 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

%p 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

%t 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 *)

%o (PARI) Vec((2+8*x-10*x^2)/(1-82*x-82*x^2+x^3) + O(x^30)) \\ _Michel Marcus_, Feb 29 2016

%o (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

%Y Cf. A051028, A051029.

%K nonn

%O 0,1

%A _Eric W. Weisstein_

%E Minor edits (g.f. and name) by _M. F. Hasler_, May 08 2016