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A255496 3rd diagonal of triangle in A255494.

%I A255496 #20 Dec 08 2021 12:06:19
%S A255496 1,38,1106,26544,567203,11179686,207768576,3692419776,63361188037,
%T A255496 1057109514902,17235551954894,275697361933728,4339725043253447,
%U A255496 67384965236252310,1034147721558836220,15711425790758327952,236612932874975360809,3536182524466029241958,52494462902614684280330
%N A255496 3rd diagonal of triangle in A255494.
%H A255496 G. C. Greubel, <a href="/A255496/b255496.txt">Table of n, a(n) for n = 0..850</a>
%H A255496 S. Falcon, <a href="http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf">On The Generating Functions of the Powers of the K-Fibonacci Numbers</a>, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
%H A255496 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (44,-649,2770,11885,-65240,-215431,-67286,139956,-23560,-2400).
%F A255496 From _G. C. Greubel_, Sep 20 2021: (Start)
%F A255496 a(n) = 12*a(n-1) + P(n+1)*A255495(n), where P(n) = A000129(n).
%F A255496 a(n) = (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n), where P(n) = A000129(n), Q(n) = A002203(n).
%F A255496 G.f.: (1 -6*x +83*x^2 -228*x^3 -84*x^4 -200*x^5)/((1+2*x)*(1-12*x)*(1 +2*x -x^2)*(1 -10*x -25*x^2)*(1 -12*x +4*x^2)*(1 -14*x -x^2)). (End)
%t A255496 a[n_]:= (12)^(n+4) -(-2)^(n+1) -2^n*LucasL[2*n+9, 2] -5^(n+4)*Fibonacci[n+5, 2] +(1/10)*Fibonacci[n+4, 2]*(Fibonacci[n+4, 2]^2 +(-1)^n);
%t A255496 Table[a[n], {n, 0, 30}] (* _G. C. Greubel_, Sep 20 2021 *)
%o A255496 (Sage)
%o A255496 def P(n): return lucas_number1(n, 2, -1)
%o A255496 def Q(n): return lucas_number2(n, 2, -1)
%o A255496 def a(n): return (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n)
%o A255496 [a(n) for n in (0..30)] # _G. C. Greubel_, Sep 20 2021
%Y A255496 Cf. A000129, A002203, A255494, A255495.
%K A255496 nonn
%O A255496 0,2
%A A255496 _N. J. A. Sloane_, Mar 06 2015
%E A255496 3 more terms. - _R. J. Mathar_, Jun 14 2015
%E A255496 Terms a(12) onward added by _G. C. Greubel_, Sep 20 2021