A131531 Period 6: repeat [0, 0, 1, 0, 0, -1].

0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0

Offset: 1

Author

Paul Curtz, Aug 26 2007

Keywords

• sign
• easy

Comments

Also: partial sums of A092220 shifted by two indices. - R. J. Mathar, Feb 08 2008

From Paul Curtz, Jun 05 2011: (Start)

The square array of this sequence in the top row and further rows defined as first differences of preceding rows starts (see A167613):

. 0, 0, 1, 0, 0, -1, ...

. 0, 1, -1, 0, -1, 1, ... = A092220,

. 1, -2, 1, -1, 2, -1, ... = A131556,

. -3, 3, -2, 3, -3 2, ...

. 6, -5, 5, -6, 5, -5, ...

. -11, 10, -11, 11, -10, 11, ...

. 21, -21, 22, -21, 21, -22, ...

. -42, 43, -43, 42, -43, 43, ...

The main diagonal in this array is A001045; the first superdiagonal is the negated elements of A001045, the second superdiagonal is A078008.

The left column of the array is basically the inverse binomial transform, (-1)^n * A024495(n), assuming offset 0.

The second column of the array is A131708 with alternating signs, and the third column is A024493 with alternating signs (both assuming offset 0). (End)

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,-1).

Crossrefs

Cf. A001045, A024493, A024495, A078008, A092220, A131556, A131708, A167613.

Programs

Magma

&cat[[0, 0, 1, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

Maple

A131531:=n->[0, 0, 1, 0, 0, -1][(n mod 6)+1]: seq(A131531(n), n=0..100); # Wesley Ivan Hurt, Jun 20 2016

Mathematica

PadRight[{}, 120, {0,0,1,0,0,-1}] (* Harvey P. Dale, Nov 11 2012 *)

PARI

a(n)=[0,0,1,0,0,-1][n%6+1] \\ Charles R Greathouse IV, Jun 01 2011

Formula

G.f.: x^3/(x+1)/(x^2-x+1). - R. J. Mathar, Nov 14 2007

a(n) = (-A057079(n+1) - (-1)^n)/3. - R. J. Mathar, Jun 13 2011

a(n) = -cos(Pi*(n-1)/3)/3 + sin(Pi*(n-1)/3)/sqrt(3) - (-1)^n/3. - R. J. Mathar, Oct 08 2011

a(n) = ( (-1)^n - (-1)^floor((n+2)/3) )/2. - Bruno Berselli, Jul 09 2013

a(n) + a(n-3) = 0 for n > 3. - Wesley Ivan Hurt, Jun 20 2016

Extensions

Edited by N. J. A. Sloane, Sep 15 2007