A099580 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1) * 4^(k-1).

0, 0, 1, 1, 9, 13, 65, 117, 441, 909, 2929, 6565, 19305, 45565, 126881, 309141, 833049, 2069613, 5467345, 13745797, 35877321, 90860509, 235418369, 598860405, 1544728185, 3940169805, 10135859761, 25896538981, 66507086889, 170093242813

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) * r^(k-1) has g.f. x^2/((1-r*x^2)*(1-x-r*x^2)) and satisfies the recurrence a(n) = a(n-1) + 2*r*a(n-2) - r*a(n-3) - r^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 (1,8,-4,-16).

Crossrefs

Cf. A006131, A097038, A099579.

Programs

Magma

[n le 4 select Floor((n-1)/2) else Self(n-1) +8*Self(n-2) -4*Self(n-3) -16*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 24 2022

Mathematica

LinearRecurrence[{1,8,-4,-16}, {0,0,1,1}, 51] (* G. C. Greubel, Jul 24 2022 *)

SageMath

@CachedFunction
def a(n): # a = A099580
    if (n<4): return (n//2)
    else: return a(n-1) +8*a(n-2) -4*a(n-3) -16*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 24 2022

Formula

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

a(n) = a(n-1) + 8*a(n-2) - 4*a(n-3) - 16*a(n-4).

From G. C. Greubel, Jul 24 2022: (Start)

a(n) = (4*(2/i)^(n-1)*ChebyshevU(n-1, i/4) - 2^n*(1-(-1)^n))/4.

E.g.f.: ( 4*exp(x/2)*sinh(sqrt(17)*x/2) - sqrt(17)*sinh(2*x) )/(2*sqrt(17)). (End)