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%I #53 Jan 01 2024 11:44:11
%S 0,8,288,9800,332928,11309768,384199200,13051463048,443365544448,
%T 15061377048200,511643454094368,17380816062160328,590436102659356800,
%U 20057446674355970888,681362750825443653408,23146276081390728245000,786292024016459316676608,26710782540478226038759688
%N X-values of solutions to the equation X*(X + 1) - 8*Y^2 = 0.
%C Equivalently, numbers k such that both k/2 and k+1 are squares. - _Karl-Heinz Hofmann_, Sep 20 2022
%H Seiichi Manyama, <a href="/A132592/b132592.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (35,-35,1).
%F a(0)=0, a(1)=8 and a(n) = 34*a(n-1) - a(n-2) + 16.
%F a(n) = (A056771(n) - 1)/2. - _Max Alekseyev_, Nov 13 2009
%F a(n) = sinh(2*n*arccosh(sqrt(2))^2) (n=0,1,2,3,...). - _Artur Jasinski_, Feb 10 2010
%F G.f.: -8*x*(x+1)/((x-1)*(x^2-34*x+1)). - _Colin Barker_, Oct 24 2012
%F a(n) = A055792(n+1)-1 = A001541(n)^2 - 1. - _Antti Karttunen_, Oct 03 2016
%t Table[Round[N[Sinh[2 n ArcCosh[Sqrt[2]]]^2, 100]], {n, 0, 20}] (* _Artur Jasinski_, Feb 10 2010 *)
%t LinearRecurrence[{35, -35, 1}, {0, 8, 288}, 30] (* _Vincenzo Librandi_, Dec 24 2018 *)
%o (Magma) I:=[0,8,288]; [n le 3 select I[n] else 35*Self(n-1)-35*Self(n-2)+ Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Dec 24 2018
%o (Python)
%o A132592 = [0, 8]
%o for n in range(2, 18): A132592.append(34 * A132592[-1] - A132592[-2] + 16)
%o print(A132592) # _Karl-Heinz Hofmann_, Sep 20 2022
%Y Intersection between A132411 and A001105.
%Y Cf. A007654.
%Y Cf. A001541, A058331, A001079, A037270, A055792, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.
%K nonn,easy
%O 0,2
%A _Mohamed Bouhamida_, Nov 14 2007
%E More terms from _Max Alekseyev_, Nov 13 2009
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