%I #27 Sep 22 2021 05:01:21
%S 1,105,8575,567203,32897774,1736613466,85474679858,3985272984490,
%T 177983686766655,7675333342669951,321533970710475033,
%U 13145650587005246037,526435406695455725140,20710119055883150135480,802278112017623387734420,30663507276425403310594244,1158197029073059563909854477
%N 4th diagonal of triangle in A255494.
%H G. C. Greubel, <a href="/A255497/b255497.txt">Table of n, a(n) for n = 0..600</a>
%H 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 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (131, -6288, 119160,-72338,-20691314,43119120,1745477304, 5765440363,-4766158745, -22941732072,25990149864,-2819340784,-2805312400,-199344000,8352000).
%F From _G. C. Greubel_, Sep 20 2021: (Start)
%F a(n) = 29*a(n-1) + P(n+1)*A255496(n).
%F a(n) = (1/7680)*( 7680*(29)^(n+5) -192*(-5)^(n+6) -30 + Q(4*n+18) -96*5^(n+6)*Q(2*n+11) +12*(-1)^n*Q(2*n+9) +3*2^(n+10)*P(3*n+15) -640*(12)^(n+6)*P(n+6) -15*(-2)^(n+10)*P(n+5) ), where P(n) = A000129(n) and Q(n) = A002203(n).
%F G.f.: (1 -26*x +1108*x^2 -15042*x^3 +74319*x^4 +67340*x^5 +1376444*x^6 +2010720*x^7 -323920*x^8 +288000*x^9)/((1-x)*(1+5*x)*(1-29*x)*(1 +4*x -4*x^2)*(1 +6*x +x^2)*(1 -24*x -144*x^2)*(1 -28*x -4*x^2)*(1 -30*x +25*x^2)*(1 -34*x +x^2)).
%F (End)
%t A255496[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 a[n_]:= a[n]= If[n<2, (105)^n, 29*a[n-1] + Fibonacci[n+1,2]*A255496[n]];
%t Table[a[n], {n,0,30}] (* _G. C. Greubel_, Sep 20 2021 *)
%o (Sage)
%o def P(n): return lucas_number1(n, 2, -1)
%o def Q(n): return lucas_number2(n, 2, -1)
%o def a(n): return (1/7680)*( 7680*(29)^(n+5) -192*(-5)^(n+6) -30 + Q(4*n+18) -96*5^(n+6)*Q(2*n+11) +12*(-1)^n*Q(2*n+9) +3*2^(n+10)*P(3*n+15) -640*(12)^(n+6)*P(n+6) -15*(-2)^(n+10)*P(n+5) )
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Sep 20 2021
%Y Cf. A000129, A002203, A255494, A255495, A255496.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Mar 06 2015
%E a(8)-a(10) from _R. J. Mathar_, Jun 14 2015
%E Terms a(11) onward added by _G. C. Greubel_, Sep 20 2021