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a(n) = 676*n^2 - 26.
2

%I #26 Jan 16 2025 11:30:16

%S 650,2678,6058,10790,16874,24310,33098,43238,54730,67574,81770,97318,

%T 114218,132470,152074,173030,195338,218998,244010,270374,298090,

%U 327158,357578,389350,422474,456950,492778,529958,568490,608374,649610,692198,736138,781430,828074

%N a(n) = 676*n^2 - 26.

%C The identity (52*n^2 - 1)^2 - (676*n^2 - 26)*(2*n)^2 = 1 can be written as A158640(n)^2 - a(n)*A005843(n)^2 = 1.

%H Vincenzo Librandi, <a href="/A158639/b158639.txt">Table of n, a(n) for n = 1..10000</a>

%H Vincenzo Librandi, <a href="https://web.archive.org/web/20090309225914/http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>, Math Forum, 2007. [Wayback Machine link]

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: 26*x*(-25 - 28*x + x^2)/(x-1)^3.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

%F From _Amiram Eldar_, Mar 19 2023: (Start)

%F Sum_{n>=1} 1/a(n) = (1 - cot(Pi/sqrt(26))*Pi/sqrt(26))/52.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/sqrt(26))*Pi/sqrt(26) - 1)/52. (End)

%F From _Elmo R. Oliveira_, Jan 16 2025: (Start)

%F E.g.f.: 26*(exp(x)*(26*x^2 + 26*x - 1) + 1).

%F a(n) = 26*A158551(n). (End)

%t LinearRecurrence[{3, -3, 1}, {650, 2678, 6058}, 50] (* _Vincenzo Librandi_, Feb 17 2012 *)

%o (Magma) I:=[650, 2678, 6058]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Feb 17 2012

%o (PARI) for(n=1, 40, print1(676*n^2 - 26", ")); \\ _Vincenzo Librandi_, Feb 17 2012

%Y Cf. A005843, A158551, A158640.

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Mar 23 2009

%E Comment rephrased and redundant formula replaced by _R. J. Mathar_, Oct 19 2009