OFFSET
0,5
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 (1,6,-3,-9).
FORMULA
G.f.: x^2/((1-3*x^2)*(1-x-3*x^2)).
a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 9*a(n-4).
From G. C. Greubel, Jul 24 2022: (Start)
a(n) = (i*sqrt(3))^(n-1)*ChebyshevU(n-1, -i/(2*sqrt(3))) - 3^((n-1)/2)*(1 - (-1)^n)/2.
E.g.f.: (1/sqrt(39))*( 2*sqrt(3)*exp(x/2)*sinh(sqrt(13)*x/2) - sqrt(13)*sinh(sqrt(3)*x) ). (End)
MATHEMATICA
LinearRecurrence[{1, 6, -3, -9}, {0, 0, 1, 1}, 50] (* G. C. Greubel, Jul 24 2022 *)
PROG
(Magma) [n le 4 select Floor((n-1)/2) else Self(n-1) +6*Self(n-2) -3*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 24 2022
(SageMath)
@CachedFunction
def a(n): # a = A099579
if (n<4): return (n//2)
else: return a(n-1) +6*a(n-2) -3*a(n-3) -9*a(n-4)
[a(n) for n in (0..40)] # G. C. Greubel, Jul 24 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 23 2004
STATUS
approved