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

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