A099858 A Chebyshev transform of (1+3x)/(1-3x).

1, 6, 17, 42, 109, 288, 755, 1974, 5167, 13530, 35423, 92736, 242785, 635622, 1664081, 4356618, 11405773, 29860704, 78176339, 204668310, 535828591, 1402817466, 3672623807, 9615053952, 25172538049, 65902560198, 172535142545

Offset: 0

Author

Paul Barry, Oct 28 2004

Keywords

• easy
• nonn

Comments

The g.f. is related to the g.f. of A099856 by the Chebyshev mapping G(x)-> (1/(1+x^2))*G(x/(1+x^2)).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,3,-1).

Crossrefs

Cf. A099856, A099857.

Programs

Mathematica

LinearRecurrence[{3,-2,3,-1},{1,6,17,42},40] (* Harvey P. Dale, Apr 17 2024 *)

Formula

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*(6*3^(n-2*k-1)-0^(n-2*k)).

a(n) = Sum_{k=0..n} (0^k+6*Fibonacci(2*k))*cos(Pi*(n-k)/2).

a(n) = Sum_{k=0..n} A099857(k)*cos(Pi*(n-k)/2).

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

a(n) = (1/2)*(4*Fibonacci(2*n+2) - i^n - (-i)^n). - Ralf Stephan, Dec 04 2004