A027003 - OEIS

A027003

a(n) = A026998(2*n, n+4).

1, 64, 901, 6821, 36425, 155793, 573382, 1899933, 5844446, 17056486, 47974934, 131553646, 354615679, 945220982, 2501450971, 6590435731, 17316698039, 45428211431, 119066290172, 311909267867, 816853717452, 2138914514428, 5600243896572, 14662288678348, 38387242941837

(list; graph; refs; listen; history; text; internal format)

OFFSET

4,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 4..1000

Index entries for linear recurrences with constant coefficients, signature (11,-53,148,-266,322,-266,148,-53,11,-1).

FORMULA

From G. C. Greubel, Jul 20 2025: (Start)

a(n) = Lucas(2*n+9) - f(n), where f(n) = (48*n^7 - 280*n^6 + 1596*n^5 - 910*n^4 + 10122*n^3 + 20405*n^2 + 46509*n + 47880)/630.

G.f.: x^4*(1 + 53*x + 250*x^2 + 154*x^3 - 59*x^4 - 15*x^5)/((1-x)^8*(1-3*x+x^2)).

E.g.f.: 4*exp(3*x/2)*(19*cosh(p*x) + 17*p*sinh(p*x)) - (1/630)*(47880 + 77490*x + 62685*x^2 + 33810*x^3 + 13650*x^4 + 4116*x^5 + 728*x^6 + 48*x^7)*exp(x), where 2*p = sqrt(5). (End)

MATHEMATICA

f[n_]:= (48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n+47880)/630;

A027003[n_]:= LucasL[2*n+9] -f[n];

Table[A027003[n], {n, 4, 50}] (* G. C. Greubel, Jul 20 2025 *)

PROG

(Magma)

A027003:= func< n | Lucas(2*n+9) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)/630 >;

[A027003(n): n in [4..45]]; // G. C. Greubel, Jul 20 2025

(SageMath)

def A027003(n): return lucas_number2(2*n+9, 1, -1) -(48*n^7 -280*n^6 +1596*n^5 -910*n^4 +10122*n^3 +20405*n^2 +46509*n +47880)//630

print([A027003(n) for n in range(4, 46)]) # G. C. Greubel, Jul 20 2025

CROSSREFS