A027018 a(n) = T(2*n+1, n+3), T given by A027011.

1, 9, 150, 1085, 5283, 20495, 69007, 212020, 613633, 1708508, 4640978, 12414802, 32903418, 86731043, 227905816, 597838223, 1566763325, 4103989113, 10747219441, 28140274566, 73676929931, 192894712070, 505012447636, 1322149114676, 3461442847524, 9062189100301

Offset: 2

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-34,71,-90,71,-34,9,-1).

Crossrefs

Cf. A000032, A027011.

Programs

Magma

A027018:= func< n | n eq 2 select 1 else Lucas(2*n+8) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 >;
[A027018(n): n in [2..50]]; // G. C. Greubel, Jun 16 2025

Mathematica

Table[LucasL[2*n+8] -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)/30 + Boole[n==2], {n,2,50}] (* G. C. Greubel, Jun 16 2025 *)

PARI

Vec(x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8)/((1-x)^6*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

SageMath

def A027018(n): return lucas_number2(2*n+8,1,-1) -(1410 +1351*n +655*n^2 +230*n^3 +20*n^4 +24*n^5)//30 + int(n==2)
print([A027018(n) for n in range(2,51)]) # G. C. Greubel, Jun 16 2025

Formula

G.f.: x^2*(1+103*x^2-30*x^3+69*x^4-73*x^5+34*x^6-9*x^7+x^8) / ((1-x)^6*(1-3*x+x^2)). - Colin Barker, Feb 19 2016

From G. C. Greubel, Jun 16 2025: (Start)

a(n) = A000032(2*n+8) - (1/30)*(1410 + 1351*n + 655*n^2 + 230*n^3 + 20*n^4 + 24*n^5) + [n=2].

E.g.f.: exp(3*x/2)*( 47*cosh(sqrt(5)*x/2) + 21*sqrt(5)*sinh(sqrt(5)*x/2) ) + x^2/2 - (1/30)*(1410 + 2280*x + 1845*x^2 + 950*x^3 + 260*x^4 + 24*x^5)*exp(x). (End)