OFFSET
0,2
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).
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)
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 06 2015
EXTENSIONS
Terms a(13) onward from G. C. Greubel, Sep 20 2021
STATUS
approved