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

1,2

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).

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)

Mathematica

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

Prog

(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
(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
(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

Crossrefs

Cf. A000032, A026998, A027963.

Sequence in context: A044651 A287303 A142216 * A387913 A061696 A062643

Adjacent sequences: A027002 A027003 A027004 * A027006 A027007 A027008

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved