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%I #44 Mar 28 2024 21:54:46
%S 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,
%T 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,
%U 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
%N Period 6: repeat [0, 0, 1, 0, 0, -1].
%C Also: partial sums of A092220 shifted by two indices. - _R. J. Mathar_, Feb 08 2008
%C From _Paul Curtz_, Jun 05 2011: (Start)
%C The square array of this sequence in the top row and further rows defined as first differences of preceding rows starts (see A167613):
%C . 0, 0, 1, 0, 0, -1, ...
%C . 0, 1, -1, 0, -1, 1, ... = A092220,
%C . 1, -2, 1, -1, 2, -1, ... = A131556,
%C . -3, 3, -2, 3, -3 2, ...
%C . 6, -5, 5, -6, 5, -5, ...
%C . -11, 10, -11, 11, -10, 11, ...
%C . 21, -21, 22, -21, 21, -22, ...
%C . -42, 43, -43, 42, -43, 43, ...
%C The main diagonal in this array is A001045; the first superdiagonal is the negated elements of A001045, the second superdiagonal is A078008.
%C The left column of the array is basically the inverse binomial transform, (-1)^n * A024495(n), assuming offset 0.
%C 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)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-1).
%F G.f.: x^3/(x+1)/(x^2-x+1). - _R. J. Mathar_, Nov 14 2007
%F a(n) = (-A057079(n+1) - (-1)^n)/3. - _R. J. Mathar_, Jun 13 2011
%F a(n) = -cos(Pi*(n-1)/3)/3 + sin(Pi*(n-1)/3)/sqrt(3) - (-1)^n/3. - _R. J. Mathar_, Oct 08 2011
%F a(n) = ( (-1)^n - (-1)^floor((n+2)/3) )/2. - _Bruno Berselli_, Jul 09 2013
%F a(n) + a(n-3) = 0 for n > 3. - _Wesley Ivan Hurt_, Jun 20 2016
%p A131531:=n->[0, 0, 1, 0, 0, -1][(n mod 6)+1]: seq(A131531(n), n=0..100); # _Wesley Ivan Hurt_, Jun 20 2016
%t PadRight[{}, 120, {0,0,1,0,0,-1}] (* _Harvey P. Dale_, Nov 11 2012 *)
%o (PARI) a(n)=[0,0,1,0,0,-1][n%6+1] \\ _Charles R Greathouse IV_, Jun 01 2011
%o (Magma) &cat[[0, 0, 1, 0, 0, -1]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016
%Y Cf. A001045, A024493, A024495, A078008, A092220, A131556, A131708, A167613.
%K sign,easy
%O 1,1
%A _Paul Curtz_, Aug 26 2007
%E Edited by _N. J. A. Sloane_, Sep 15 2007
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