A117373 Expansion of (1 - 3x)/(1 - x + x^2).

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

Offset: 0

Author

Paul Barry, Mar 10 2006

Keywords

• easy
• sign

Comments

Row sums of number triangle A117372.

Periodic sequence with period {1, -2, -3, -1, 2, 3}. - Philippe Deléham, Nov 03 2008

Links

• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,-1).

Crossrefs

Cf. A010892, A117372.

Cf. A010872 (n mod 3), A010875 (n mod 6).

Programs

Mathematica

CoefficientList[Series[(1 - 3 x)/(1 - x + x^2), {x, 0, 200}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)

PARI

Vec((1-3*x)/(1-x+x^2) + O(x^90)) \\ Michel Marcus, Apr 06 2016

Formula

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

a(n) = Sum_{k=0..n} (-1)^(n-k)*(C(k,n-k) + 3*C(k, n-k-1)).

a(n) = a(n-1) - a(n-2); a(0)=1, a(1)=-2. - Philippe Deléham, Nov 03 2008

a(n) = A010892(n) - 3*A010892(n-1). - R. J. Mathar, Sep 14 2013

a(n) = cos(n*Pi/3) - 5*sin(n*Pi/3)/sqrt(3). - Andres Cicuttin, Apr 06 2016

a(n) = ((n mod 3)^2 - 4*(n mod 3) + 1)*(-1)^floor(n/3). - Luce ETIENNE, Nov 18 2017