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A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.

%I A182191 #36 Sep 08 2022 08:45:55
%S A182191 -1,5,43,265,1559,9101,53059,309265,1802543,10506005,61233499,
%T A182191 356895001,2080136519,12123924125,70663408243,411856525345,
%U A182191 2400475743839,13990997937701,81545511882379,475282073356585,2770146928257143,16145599496186285,94103450048860579
%N A182191 a(n) = 6*a(n-1) - a(n-2) + 12 with n>1, a(0)=-1, a(1)=5.
%C A182191 If p is a prime of the form 8*r +/- 3 then a(p) == 1 (mod p); if p is a prime of the form 8*r +/- 1 then a(p) == 5 (mod p).
%H A182191 Bruno Berselli, <a href="/A182191/b182191.txt">Table of n, a(n) for n = 0..100</a>
%H A182191 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,-7,1).
%F A182191 G.f.: -(1-12*x-x^2)/((1-x)*(1-6*x+x^2)). - _Bruno Berselli_, May 15 2012
%F A182191 a(n) = 2*A038723(n) - 3. -_Bruno Berselli_, May 16 2012
%F A182191 a(n) = -3 + (1/4)*( (4+sqrt(2))*(3+2*sqrt(2))^n + (4-sqrt(2))*(3-2*sqrt(2))^n ). - _Colin Barker_, Mar 05 2016
%F A182191 From _G. C. Greubel_, May 24 2021: (Start)
%F A182191 a(n) = A000129(2*n+2) - 2*A000129(2*n) - 3.
%F A182191 a(n) = A000129(2*n) + A002203(2*n) - 3. (End)
%t A182191 m = 19;n = 1; c = 0;
%t A182191 list3 = Reap[While[c < 22, t = 6 n - m + 12; Sow[t];m = n; n = t;c++]][[2,1]]
%t A182191 Table[LucasL[2*n, 2] +Fibonacci[2*n, 2] -3, {n, 0, 40}] (* _G. C. Greubel_, May 24 2021 *)
%o A182191 (Magma) I:=[-1,5]; [n le 2 select I[n] else 6*Self(n-1)-Self(n-2)+12: n in [1..19]]; // _Bruno Berselli_, May 15 2012
%o A182191 (Sage) [lucas_number1(2*n+2,2,-1) - 2*lucas_number1(2*n,2,-1) -3 for n in (0..40)] # _G. C. Greubel_, May 24 2021
%Y A182191 Cf. A000129, A002203, A038723.
%K A182191 sign,easy
%O A182191 0,2
%A A182191 _Kenneth J Ramsey_, Apr 17 2012