A027009 a(n) = Sum_{k=floor((n+2)/2)..n} T(n, k), T given by A026998.

1, 1, 9, 14, 46, 81, 209, 389, 901, 1726, 3774, 7349, 15541, 30561, 63329, 125294, 256366, 509161, 1033449, 2057549, 4154701, 8284926, 16673534, 33282989, 66837421, 133507081, 267724809, 535010414, 1071881326, 2142612801, 4290096449, 8577182549, 17167117141, 34326353086, 68686091454, 137351549669, 274790503141

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-1,2).

Formula

G.f.: x*(1-x+4*x^2-x^3-2*x^4)/((1-2*x)*(1+x-x^2)*(1-x-x^2)).

From G. C. Greubel, Jul 11 2025: (Start)

a(n) = 2^(n+1) - (1/2)*(A000032(n+3) + (-1)^n*A000032(n)).

E.g.f.: 1 + 2*exp(2*x) - 2*cosh(x/2)*cosh(sqrt(5)*x/2) - exp(x/2)*(cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2)). (End)

Mathematica

LinearRecurrence[{2, 3, -6, -1, 2}, {1, 1, 9, 14, 46}, 41] (* G. C. Greubel, Jul 11 2025 *)

Prog

(Magma)
A027009:= func< n | 2^(n+1) -(Lucas(n+3) +(-1)^n*Lucas(n))/2 >;
[A027009(n): n in [1..40]]; // G. C. Greubel, Jul 11 2025
(SageMath)
def lucas(n): return lucas_number2(n, 1, -1)
def A027009(n): return 2^(n+1) -(lucas(n+3) +(-1)^n*lucas(n))//2
print([A027009(n) for n in range(1, 41)]) # G. C. Greubel, Jul 11 2025
(PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 2, -1, -6, 3, 2]^(n-1)*[1; 1; 9; 14; 46])[1, 1] \\ Charles R Greathouse IV, Jun 02 2026

Crossrefs

Cf. A000032, A026998.

Sequence in context: A139055 A294030 A079625 * A239038 A271596 A272275

Adjacent sequences: A027006 A027007 A027008 * A027010 A027011 A027012

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

More terms added by G. C. Greubel, Jul 11 2025

Status

approved