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

0,2

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).

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)

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A199353 A263888 A269474 * A094075 A199519 A082368

Adjacent sequences: A255494 A255495 A255496 * A255498 A255499 A255500

Keyword

nonn

Author

N. J. A. Sloane, Mar 06 2015

Extensions

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

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

Status

approved