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A157267 a(n) = 10368*n^2 - 4896*n + 577.

%I A157267 #30 Nov 13 2024 06:57:57
%S A157267 6049,32257,79201,146881,235297,344449,474337,624961,796321,988417,
%T A157267 1201249,1434817,1689121,1964161,2259937,2576449,2913697,3271681,
%U A157267 3650401,4049857,4470049,4910977,5372641,5855041,6358177,6882049,7426657,7992001,8578081,9184897,9812449
%N A157267 a(n) = 10368*n^2 - 4896*n + 577.
%C A157267 The identity (10368*n^2 - 4896*n + 577)^2 - (36*n^2 - 17*n + 2)*(1728*n - 408)^2 = 1 can be written as a(n)^2 - A157265(n)*A157266(n)^2 = 1. - _Vincenzo Librandi_, Jan 27 2012
%C A157267 This is the case s = 4*n - 1 of the identity (2*r^2 - 1)^2 - ((r^2 - 1)/144)*(24*r)^2 = 1, where r = 18*s + 9*i^(s*(s+1)) - (-1)^s - 9 and i = sqrt(-1). - _Bruno Berselli_, Jan 29 2012
%H A157267 Vincenzo Librandi, <a href="/A157267/b157267.txt">Table of n, a(n) for n = 1..10000</a>
%H A157267 Vincenzo Librandi, <a href="http://mathforum.org/kb/message.jspa?messageID=5785989&amp;tstart=0">X^2-AY^2=1</a>.
%H A157267 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A157267 From _Vincenzo Librandi_, Jan 27 2012: (Start)
%F A157267 G.f.: x*(6049 + 14110*x + 577*x^2)/(1-x)^3.
%F A157267 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A157267 E.g.f.: exp(x)*(10368*x^2 + 5472*x + 577) - 577. - _Elmo R. Oliveira_, Nov 09 2024
%t A157267 LinearRecurrence[{3,-3,1},{6049,32257,79201},40] (* _Vincenzo Librandi_, Jan 27 2012 *)
%o A157267 (Magma) I:=[6049, 32257, 79201]; [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_, Jan 27 2012
%o A157267 (PARI) for(n=1, 40, print1(10368*n^2 - 4896*n + 577", ")); \\ _Vincenzo Librandi_, Jan 27 2012
%Y A157267 Cf. A157265, A157266.
%K A157267 nonn,easy
%O A157267 1,1
%A A157267 _Vincenzo Librandi_, Feb 26 2009