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%I #18 Sep 08 2022 08:45:16
%S 1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,
%T 0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,
%U 0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0,1,1,-1,-1,0,0
%N Period 6: repeat [1, 1, -1, -1, 0, 0].
%C The positive sequence is A131719(n+1) = a(n) = (cos(2*Pi*n/3+Pi/3)/6+sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 -sqrt(3)*cos(Pi*n/3+Pi/6)/6+sin(Pi*n/3+Pi/6)/2+2/3, with g.f. (1+x^2) / ( (1-x)*(1-x+x^2)*(1+x+x^2) ).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,-1).
%F G.f.: (1+x)/(1+x^2+x^4).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(k, floor(n/2)-k)*(-1)^k.
%F a(n) = -cos(2*Pi*n/3+Pi/3)/2 + sqrt(3)*sin(2*Pi*n/3+Pi/3)/6 + sqrt(3)*cos(Pi*n/3+Pi/6)/2 + sin(Pi*n/3+Pi/6)/2.
%F a(n) = cos(Pi*n/3) + sin(2*Pi*n/3)/sqrt(3). - _R. J. Mathar_, Oct 08 2011
%F a(n) + a(n-2) + a(n-4) = 0 for n>3. - _Wesley Ivan Hurt_, Jun 20 2016
%F E.g.f.: (sqrt(3)*sin(sqrt(3)*x/2) + 3*cos(sqrt(3)*x/2)*exp(x))*exp(-x/2)/3. - _Ilya Gutkovskiy_, Jun 21 2016
%p A103368:=n->[1, 1, -1, -1, 0, 0][(n mod 6)+1]: seq(A103368(n), n=0..100); # _Wesley Ivan Hurt_, Jun 20 2016
%t PadRight[{}, 100, {1, 1, -1, -1, 0, 0}] (* _Wesley Ivan Hurt_, Jun 20 2016 *)
%o (Magma) &cat [[1, 1, -1, -1, 0, 0]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016
%Y Cf. A131719.
%K easy,sign
%O 0,1
%A _Paul Barry_, Feb 02 2005
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