A027006 a(n) = T(2*n+1, n+3), T given by A026998.

1, 34, 281, 1397, 5353, 17643, 53062, 150833, 414210, 1114160, 2960806, 7814074, 20544191, 53902532, 141273663, 370060623, 969088727, 2537431693, 6643486220, 17393369595, 45537037936, 119218243314, 312118286876, 817137321092, 2139294503373, 5600747154678

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Bisection of A053298.

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-26,45,-45,26,-8,1).

Crossrefs

Cf. A000032, A026998, A053298.

Programs

Magma

A027006:= func< n | Lucas(2*n+7) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6 >;
[A027006(n): n in [2..40]]; // G. C. Greubel, Jul 22 2025

Mathematica

A027006[n_]:= LucasL[2*n+7] -(12*n^4 +20*n^3 +81*n^2 +169*n +174)/6;
Table[A027006[n], {n,2,42}] (* G. C. Greubel, Jul 22 2025 *)

PARI

Vec(x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

SageMath

def A027006(n): return lucas_number2(2*n+7,1,-1) -(12*n^4 +20*n^3 +81*n^2 +169*n +174)//6
print([A027006(n) for n in range(2,41)]) # G. C. Greubel, Jul 22 2025

Formula

G.f.: x^2*(1+26*x+35*x^2-12*x^3-2*x^4) / ((1-x)^5*(1-3*x+x^2)). - Colin Barker, Feb 19 2016

From G. C. Greubel, Jul 22 2025: (Start)

a(n) = Lucas(2*n+7) - (12*n^4 + 20*n^3 + 81*n^2 + 169*n + 174)/6.

E.g.f.: exp(3*x/2)*(29*cosh(p*x) + 26*p*sinh(p*x)) - (1/6)*(174 + 282*x + 225*x^2 + 92*x^3 + 12*x^4)*exp(x), where 2*p = sqrt(5). (End)