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%I #56 Jan 31 2024 08:31:52
%S 0,4,28,168,984,5740,33460,195024,1136688,6625108,38613964,225058680,
%T 1311738120,7645370044,44560482148,259717522848,1513744654944,
%U 8822750406820,51422757785980,299713796309064
%N a(n) = 6*a(n-1) - a(n-2) + 4 with n > 1, a(0)=0, a(1)=4.
%C Also, nonnegative m such that 2m(m+2)+1 is a square. - _Bruno Berselli_, Oct 22 2012
%H G. C. Greubel, <a href="/A182190/b182190.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F G.f.: 4*x/((1-x)*(1-6*x+x^2)). - _Bruno Berselli_, Jun 07 2012
%F a(n) = 4*A053142(n). - _Bruno Berselli_, Jun 07 2012
%F a(n) = A001653(n+1) - 1. - _Kiran S. Kedlaya_, Mar 14 2021
%F a(n) = A000129(2*n+1) - 1. - _G. C. Greubel_, May 24 2021
%F a(n) = A143608(n)*A143608(n+1). - _R. J. Mathar_, Jan 31 2024
%F a(n)-a(n-1) = A005319(n). - _R. J. Mathar_, Jan 31 2024
%t m = 4;n = 0; c = 0;
%t list3 = Reap[While[c < 22, t = 6 n - m + 4; Sow[t];m = n; n = t;c++]][[2,1]]
%t Table[Fibonacci[2*n+1, 2] -1, {n,0,40}] (* _G. C. Greubel_, May 24 2021 *)
%o (Magma) I:=[0,4]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+4: n in [1..20]]; // _Bruno Berselli_, Jun 07 2012
%o (Sage) [lucas_number1(2*n+1,2,-1) -1 for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y Cf. A000129, A182189, A182431.
%K nonn,easy
%O 0,2
%A _Kenneth J Ramsey_, Apr 17 2012
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