This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051030 #56 Feb 16 2025 08:32:41
%S A051030 2,172,14258,1183258,98196140,8149096378,676276803218,56122825570732,
%T A051030 4657518245567522,386517891556533610,32076327480946722092,
%U A051030 2661948663027021400042,220909662703761829481378,18332840055749204825554348,1521404814964480238691529490
%N A051030 Ramanujan's c-series: expansion of (2+8*x-10*x^2)/(1-82*x-82*x^2+x^3).
%C A051030 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 A051030 Harvey P. Dale, <a href="/A051030/b051030.txt">Table of n, a(n) for n = 0..500</a>
%H A051030 Kwang-Wu Chen, <a href="https://web.archive.org/web/2024*/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 A051030 Jung Hun Han and Michael 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 A051030 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 A051030 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 A051030 J. Mc Laughlin, <a href="https://web.archive.org/web/2024*/https://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 A051030 Eric Rowland and Jesus Sistos Barron, <a href="https://arxiv.org/abs/2501.14643">Complexity of powers of a constant-recursive sequence</a>, arXiv:2501.14643 [math.NT], 2025. See p. 2.
%H A051030 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujansSumIdentity.html">Ramanujan's Sum Identity</a>.
%H A051030 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (82,82,-1).
%F A051030 G.f.: (2+8*x-10*x^2)/((1+x)*(1-83*x+x^2)).
%F A051030 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
%F A051030 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 A051030 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 A051030 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 A051030 (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 A051030 (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 A051030 Cf. A051028, A051029.
%K A051030 nonn,easy
%O A051030 0,1
%A A051030 _Eric W. Weisstein_
%E A051030 Minor edits (g.f. and name) by _M. F. Hasler_, May 08 2016