A100219 Expansion of (1-2*x)/((1-x)*(1-x+x^2)).

1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0, 1, 0, -2, -3, -2, 0

Offset: 0

Author

Paul Barry, Nov 08 2004

Keywords

• easy
• sign

Comments

Row sums of number triangle A100218.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Crossrefs

Cf. A079757, A100218.

Programs

Magma

&cat[[1,0,-2,-3,-2,0]: n in [0..20]]; // G. C. Greubel, Mar 28 2024

Mathematica

PadRight[{}, 120, {1,0,-2,-3,-2,0}] (* or *) LinearRecurrence[{2,-2,1}, {1,0,-2}, 50] (* G. C. Greubel, Mar 13 2017; Mar 28 2024 *)
Table[Cos[Pi*n/3 + Pi/3] + Sqrt[3]*Sin[Pi*n/3 + Pi/3] - 1, {n, 0, 71}] (* Indranil Ghosh, Mar 13 2017 *)

PARI

my(x='x+O('x^50)); Vec((1-2*x)/((1-x)*(1-x+x^2))) \\ G. C. Greubel, Mar 13 2017

SageMath

def A100219(n): return [1,0,-2,-3,-2,0][n%6]
[A100219(n) for n in range(121)] # G. C. Greubel, Mar 28 2024

Formula

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

a(n) = cos(Pi*n/3 + Pi/3) + sqrt(3)*sin(Pi*n/3 + Pi/3) - 1.

a(n) is the n-th order Taylor polynomial (centered at 0) of 1/c(x)^n evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. - Peter Bala, Apr 20 2024