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A131531
Period 6: repeat [0, 0, 1, 0, 0, -1].
18
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
1,1
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)
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
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 *)
PROG
(PARI) a(n)=[0, 0, 1, 0, 0, -1][n%6+1] \\ Charles R Greathouse IV, Jun 01 2011
(Magma) &cat[[0, 0, 1, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Aug 26 2007
EXTENSIONS
Edited by N. J. A. Sloane, Sep 15 2007
STATUS
approved