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A158657 a(n) = 784*n^2 - 28.

%I A158657 #25 Jan 16 2025 12:28:39
%S A158657 756,3108,7028,12516,19572,28196,38388,50148,63476,78372,94836,112868,
%T A158657 132468,153636,176372,200676,226548,253988,282996,313572,345716,
%U A158657 379428,414708,451556,489972,529956,571508,614628,659316,705572,753396,802788,853748,906276,960372
%N A158657 a(n) = 784*n^2 - 28.
%C A158657 The identity (56*n^2 - 1)^2 - (784*n^2 - 28)*(2*n)^2 = 1 can be written as A158658(n)^2 - a(n)*A005843(n)^2 = 1.
%H A158657 Vincenzo Librandi, <a href="/A158657/b158657.txt">Table of n, a(n) for n = 1..10000</a>
%H A158657 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 A158657 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A158657 G.f.: 28*x*(-27 - 30*x + x^2)/(x-1)^3.
%F A158657 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A158657 From _Amiram Eldar_, Mar 20 2023: (Start)
%F A158657 Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)))/56.
%F A158657 Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(7)))*Pi/(2*sqrt(7)) - 1)/56. (End)
%F A158657 From _Elmo R. Oliveira_, Jan 16 2025: (Start)
%F A158657 E.g.f.: 28*(exp(x)*(28*x^2 + 28*x - 1) + 1).
%F A158657 a(n) = 28*A158554(n). (End)
%t A158657 LinearRecurrence[{3, -3, 1}, {756, 3108, 7028}, 50] (* _Vincenzo Librandi_, Feb 17 2012 *)
%o A158657 (Magma) I:=[756, 3108, 7028]; [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 A158657 (PARI) for(n=1, 40, print1(784*n^2 - 28", ")); \\ _Vincenzo Librandi_, Feb 17 2012
%Y A158657 Cf. A005843, A158554, A158658.
%K A158657 nonn,easy
%O A158657 1,1
%A A158657 _Vincenzo Librandi_, Mar 23 2009
%E A158657 Comment rephrased and redundant formula replaced by _R. J. Mathar_, Oct 19 2009