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a(n) = -cos((2 n - 1) arcsin(sqrt(3)))^2 = -1 + cosh((2 n - 1) arcsinh(sqrt(2)))^2.
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%I #31 Jun 10 2018 20:31:32

%S 2,242,23762,2328482,228167522,22358088722,2190864527282,

%T 214682365584962,21036680962799042,2061380051988721202,

%U 201994208413931878802,19793371044513335401442,1939548368153892937462562,190055946708036994535929682,18623543229019471571583646322

%N a(n) = -cos((2 n - 1) arcsin(sqrt(3)))^2 = -1 + cosh((2 n - 1) arcsinh(sqrt(2)))^2.

%H G. C. Greubel, <a href="/A146312/b146312.txt">Table of n, a(n) for n = 1..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (99,-99,1).

%F General formula: cosh((2*n-1)*arcsinh(sqrt(2)))^2 + cos((2*n-1)*arcsin(sqrt(3))^2 = 1.

%F a(n) = A146313(n) - 1.

%F a(n) = 99*a(n-1) - 99*a(n-2) + a(n-3). - _Colin Barker_, Oct 26 2014

%F G.f.: -2*x*(x^2+22*x+1) / ((x-1)*(x^2-98*x+1)). - _Colin Barker_, Oct 26 2014

%F a(n) = 2*A054320(n-1)^2. - _Jon E. Schoenfield_, Jun 08 2018

%t Table[Round[ -N[Cos[(2 n - 1) ArcSin[Sqrt[3]]], 300]^2], {n, 1, 50}]

%t LinearRecurrence[{99, -99, 1}, {2, 242, 23762}, 50] (* _G. C. Greubel_, Jul 03 2017 *)

%o (PARI) Vec(-2*x*(x^2+22*x+1) / ((x-1)*(x^2-98*x+1)) + O(x^100)) \\ _Colin Barker_, Oct 26 2014

%Y Cf. A146311, A146313.

%K nonn,easy

%O 1,1

%A _Artur Jasinski_, Oct 29 2008

%E More terms from _Colin Barker_, Oct 26 2014