A255496 3rd diagonal of triangle in A255494.

1, 38, 1106, 26544, 567203, 11179686, 207768576, 3692419776, 63361188037, 1057109514902, 17235551954894, 275697361933728, 4339725043253447, 67384965236252310, 1034147721558836220, 15711425790758327952, 236612932874975360809, 3536182524466029241958, 52494462902614684280330

Offset: 0

Author

N. J. A. Sloane, Mar 06 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..850
• S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
• Index entries for linear recurrences with constant coefficients, signature (44,-649,2770,11885,-65240,-215431,-67286,139956,-23560,-2400).

Crossrefs

Cf. A000129, A002203, A255494, A255495.

Programs

Mathematica

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);
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Sep 20 2021 *)

Sage

def P(n): return lucas_number1(n, 2, -1)
def Q(n): return lucas_number2(n, 2, -1)
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)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

Formula

From G. C. Greubel, Sep 20 2021: (Start)

a(n) = 12*a(n-1) + P(n+1)*A255495(n), where P(n) = A000129(n).

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).

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)

Extensions

3 more terms. - R. J. Mathar, Jun 14 2015

Terms a(12) onward added by G. C. Greubel, Sep 20 2021