A027005 a(n) = T(2*n+1,n+2), T given by A026998.

1, 19, 101, 370, 1148, 3278, 8967, 23993, 63483, 167040, 438346, 1148844, 3009181, 7879855, 20631713, 54016798, 141420392, 370246298, 969320643, 2537718005, 6643835991, 17393792844, 45537545686, 119218847640, 312119000953, 817138159243, 2139295481117

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Bisection of A027963.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-13,13,-6,1).

Crossrefs

Cf. A000032, A026998, A027963.

Programs

Magma

A027005:= func< n | Lucas(2*n+5) -(6*n^2+11*n+11) >;
[A027005(n): n in [1..40]]; // G. C. Greubel, Jul 21 2025

Mathematica

LinearRecurrence[{6,-13,13,-6,1},{1,19,101,370,1148},30] (* Harvey P. Dale, Aug 19 2020 *)

PARI

Vec(x*(1+13*x-2*x^3)/((1-x)^3*(1-3*x+x^2)) + O(x^40)) \\ Colin Barker, Feb 19 2016

SageMath

def A027005(n): return lucas_number2(2*n+5,1,-1) -(6*n**2 +11*n +11)
print([A027005(n) for n in range(1,41)]) # G. C. Greubel, Jul 21 2025

Formula

From Colin Barker, Feb 19 2016: (Start)

a(n) = 2^(-1-n)*((25+11*sqrt(5))*(3+sqrt(5))^n - (25-11*sqrt(5))*(3-sqrt(5))^n )/sqrt(5) + 7*(1+n) - 6*(n+1)*(n+2) + 7*(n+1) - 6.

a(n) = 6*a(n-1) - 13*a(n-2) + 13*a(n-3) - 6*a(n-4) + a(n-5) for n>5.

G.f.: x*(1+13*x-2*x^3) / ((1-x)^3*(1-3*x+x^2)). (End)

From G. C. Greubel, Jul 21 2025: (Start)

a(n) = Lucas(2*n+5) - (6*n^2 + 11*n + 11).

E.g.f.: exp(3*x/2)*(11*cosh(p*x) + 10*p*sinh(p*x)) - (11 + 17*x + 6*x^2)*exp(x), where 2*p = sqrt(5). (End)