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

Author

Clark Kimberling

Keywords

• nonn

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).

Crossrefs

Cf. A000032, A026998.

Programs

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

Mathematica

LinearRecurrence[{2,3,-6,-1,2}, {1,1,9,14,46}, 41] (* 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

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)

Extensions

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