A131531 - OEIS

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

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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)

LINKS

Table of n, a(n) for n=1..92.

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

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