A027007 a(n) = A026998(2n+1, n+4).

1, 53, 634, 4201, 20120, 78753, 269829, 844702, 2486178, 7017354, 19260116, 51903794, 138254821, 365619439, 962704734, 2528441803, 6631057180, 17376467099, 45514392201, 119188310928, 312079208726, 817086876180, 2139230058328, 5600665608772, 14662845807193

Offset: 3

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-43,105,-161,161,-105,43,-10,1).

Crossrefs

Cf. A000032, A026998.

Programs

Magma

A027007:= func< n | Lucas(2*n+9) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280) >;
[A027007(n): n in [3..45]]; // G. C. Greubel, Jul 23 2025

Mathematica

Table[LucasL[2*n+9] -(1/30)*(8*n^6 -4*n^5 +110*n^4 +325*n^3 +1052*n^2 +2199*n +2280), {n,3,45}] (* G. C. Greubel, Jul 23 2025 *)

SageMath

def A027007(n): return lucas_number2(2*n+9,1,-1) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280)
print([A027007(n) for n in range(3,46)]) # G. C. Greubel, Jul 23 2025

Formula

From G. C. Greubel, Jul 23 2025: (Start)

a(n) = Lucas(2*n+9) - (1/30)*(8*n^6 - 4*n^5 + 110*n^4 + 325*n^3 + 1052*n^2 + 2199*n + 2280).

G.f.: x^3*(1 + 43*x + 147*x^2 + 35*x^3 - 32*x^4 - 2*x^5)/((1-x)^7*(1-3*x+x^2)).

E.g.f.: 4*exp(3*x/2)*( 19*cosh(p*x) + 17*p*sinh(p*x) ) - (1/30)*(2280 + 3690*x + 2985*x^2 + 1605*x^3 + 590*x^4 + 116*x^5 + 8*x^6)*exp(x), where 2*p = sqrt(5). (End)

Extensions

More terms from Sean A. Irvine, Oct 21 2019