A100049 A Chebyshev transform of the Padovan numbers.

1, 0, -1, 1, -1, -3, 3, 3, -6, 2, 10, -13, -9, 29, -9, -43, 55, 32, -126, 48, 183, -243, -121, 541, -241, -765, 1082, 450, -2326, 1171, 3179, -4803, -1617, 9993, -5601, -13168, 21250, 5552, -42849, 26489, 54351, -93763, -17765, 183347, -124086, -223422, 412698, 49827, -782881, 576541, 914279

Offset: 0

Author

Paul Barry, Oct 31 2004

Keywords

• easy
• sign

Comments

A Chebyshev transform of the Padovan numbers A000931(n+3): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,-2,1,-2,0,-1).

Crossrefs

Cf. A099443, A011655, A100047, A100048.

Programs

Mathematica

LinearRecurrence[{0, -2, 1, -2, 0, -1}, {1, 0, -1, 1, -1, -3, 3}, 50] (* G. C. Greubel, Aug 08 2017 *)

PARI

x='x+O('x^50); Vec((1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6)) \\ G. C. Greubel, Aug 08 2017

Formula

G.f.: (1-x^2)*(1+x^2)^2/(1+2*x^2-x^3+2*x^4+x^6).

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

a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A000931(n-2*k+3)/(n-k).