A099583 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*2^(n-k-1)*(3/2)^(k-1).

0, 0, 1, 2, 10, 26, 91, 260, 820, 2420, 7381, 22022, 66430, 198926, 597871, 1792520, 5380840, 16139240, 48427561, 145272842, 435848050, 1307514626, 3922632451, 11767808780, 35303692060, 105910810460, 317733228541, 953198888462

Offset: 0

Author

Paul Barry, Oct 23 2004

Keywords

• easy
• nonn

Comments

In general, a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1)*u^(n-k-1)*(v/u)^(k-1) has g.f. x^2/((1-v*x^2)(1-u*x-v*x^2)) and satisfies the recurrence a(n) = u*a(n-1) + 2v*a(n-2) - u*v*a(n-3) - v^2*a(n-4).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,6,-6,-9).

Crossrefs

Cf. A006100, A002452.

Programs

Magma

I:=[0,0,1,2]; [n le 4 select I[n] else 2*Self(n-1) +6*Self(n-2) -6*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 22 2022

Mathematica

LinearRecurrence[{2,6,-6,-9}, {0,0,1,2}, 40] (* G. C. Greubel, Jul 22 2022 *)

PARI

a(n) = sum(k=0, n\2, binomial(n-k, k-1)*2^(n-k-1)*(3/2)^(k-1)); \\ Michel Marcus, Jan 20 2018

SageMath

[(1/8)*(3^n -(-1)^n -2*(1-(-1)^n)*3^((n-1)/2)) for n in (0..40)] # G. C. Greubel, Jul 22 2022

Formula

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

a(n) = 2*a(n-1) + 6*a(n-2) - 6*a(n-3) - 9*a(n-4).

a(n) = A002452(n/2) if n even; a(n) = 2*A006100((n+1)/2) if n odd. - R. J. Mathar, Jun 06 2010

a(0)=0, a(1)=0; a(2)=1; a(n) = 2*a(n-1) + 3*a(n-2) if n is odd; a(n) = 2*a(n-1) + 3*a(n-2) + 3^m (m=1,2,3...) if n is even. - Vincenzo Librandi, Jun 26 2010

From G. C. Greubel, Jul 22 2022: (Start)

a(n) = (1/8)*(3^n - (-1)^n - 2*(1-(-1)^n)*3^((n-1)/2) ).

E.g.f.: (1/12)*(3*exp(x)*sinh(2*x) - 2*sqrt(3)*sinh(sqrt(3)*x)). (End)