A255495 2nd diagonal of triangle in A255494.

1, 13, 130, 1106, 8575, 62475, 435576, 2939208, 19342285, 124800361, 792586270, 4969028750, 30822650251, 189500937303, 1156406300340, 7012380492516, 42294614785465, 253926386816725, 1518506730836026, 9050029200532298, 53778595325886295, 318762380704793571, 1885254096749834160

Offset: 0

Author

N. J. A. Sloane, Mar 06 2015

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• 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 (14,-56,14,189,84,-20).

Crossrefs

Cf. A000129, A094706, A255494.

Programs

Magma

I:=[1,13,130,1106,8575,62475]; [n le 6 select I[n] else 14*Self(n-1) - 56*Self(n-2) +14*Self(n-3) +189*Self(n-4) + 84*Self(n-5) -20*Self(n-6): n in [1..31]]; // G. C. Greubel, Sep 20 2021

Mathematica

a[n_]:= (1/2)*(Fibonacci[n+3, 2]*Fibonacci[n+4, 2] + 2^(n+4)*Fibonacci[n+4, 2] - 2*5^(n+3));
Table[a[n], {n,0,30}] (* G. C. Greubel, Sep 20 2021 *)

Sage

def P(n): return lucas_number1(n, 2, -1)
def A255495(n): return (1/2)*(P(n+3)*P(n+4) + 2^(n+4)*P(n+4) - 2*5^(n+3))
[A255495(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

Formula

G.f.: (1 -x +4*x^2)/((1+x)*(1-5*x)*(1-6*x+x^2)*(1-4*x-4*x^2)). - R. J. Mathar, Jun 14 2015

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

a(n) = (1/2)*(P(n+3)*P(n+4) + 2^(n+4)*P(n+4) - 2*5^(n+3)), where P(n) = A000129(n).

a(n) = 5*a(n-1) + P(n+1)*(P(n+3) - 2^(n+2)) = 5*a(n) + P(n+1)*A094706(n+1). (End)

Extensions

Terms a(13) onward from G. C. Greubel, Sep 20 2021