A255497 4th diagonal of triangle in A255494.

1, 105, 8575, 567203, 32897774, 1736613466, 85474679858, 3985272984490, 177983686766655, 7675333342669951, 321533970710475033, 13145650587005246037, 526435406695455725140, 20710119055883150135480, 802278112017623387734420, 30663507276425403310594244, 1158197029073059563909854477

Offset: 0

Author

N. J. A. Sloane, Mar 06 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..600
• 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 (131, -6288, 119160,-72338,-20691314,43119120,1745477304, 5765440363,-4766158745, -22941732072,25990149864,-2819340784,-2805312400,-199344000,8352000).

Crossrefs

Cf. A000129, A002203, A255494, A255495, A255496.

Programs

Mathematica

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);
a[n_]:= a[n]= If[n<2, (105)^n, 29*a[n-1] + Fibonacci[n+1,2]*A255496[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 (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) )
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

Formula

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

a(n) = 29*a(n-1) + P(n+1)*A255496(n).

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).

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)).

(End)

Extensions

a(8)-a(10) from R. J. Mathar, Jun 14 2015

Terms a(11) onward added by G. C. Greubel, Sep 20 2021