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%I #27 Dec 12 2023 08:42:08
%S 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,-1,0,0,-1,
%T 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,-1,0,0,
%U -1,1,1,-1,0,0,-1,1,1,-1,0,0,-1,1,1,-1,0,0,-1,1
%N Period 6: repeat [1, -1, 0, 0, -1, 1].
%C The Fi2 sums, see A180662, of triangle A108299 equal the terms of this sequence. - _Johannes W. Meijer_, Aug 11 2011
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,-1).
%F First differences of A134667.
%F Euler transform of length 6 sequence [-1, 0, 0, -1, 0, 1]. - _Michael Somos_, Feb 08 2008
%F a(n) = a(-1-n) for all n in Z. - _Michael Somos_, Feb 08 2008
%F G.f.: (1-x)*(1-x^4) / (1-x^6) = (1-x)*(1+x^2) / ((1-x+x^2)*(1+x+x^2)) = (1-x+x^2-x^3) / (1+x^2+x^4).
%F a(6*n + 2) = a(6*n + 3) = 0. - _Michael Somos_, Oct 16 2015
%F From _Wesley Ivan Hurt_, Jun 20 2016: (Start)
%F a(n) + a(n-2) + a(n-4) = 0 for n>3.
%F a(n) = cos(n*Pi/6) * (3*cos(n*Pi/2) + 2*sqrt(3)*sin(n*Pi/6) - 3*sqrt(3)*sin(n*Pi/2))/3. (End)
%e G.f. = 1 - x - x^4 + x^5 + x^6 - x^7 - x^10 + x^11 + x^12 - x^13 - x^16 + ...
%p A134668 :=proc(n): (1/6)*(-2*((n+1) mod 6)+((n+2) mod 6)-((n+4) mod 6)+2*((n+5) mod 6)) end: seq(A134668(n), n=0..74); # _Johannes W. Meijer_, Aug 14 2011
%t PadRight[{},120,{1,-1,0,0,-1,1}] (* or *) LinearRecurrence[{0,-1,0,-1},{1,-1,0,0},120] (* _Harvey P. Dale_, Dec 03 2012 *)
%o (PARI) {a(n)=[1, -1, 0, 0, -1, 1][n%6+1]}; /* _Michael Somos_, Feb 08 2008 */
%o (Magma) &cat [[1, -1, 0, 0, -1, 1]^^20]; // _Wesley Ivan Hurt_, Jun 20 2016
%Y Cf. A108299, A134667, A180662.
%K sign,easy
%O 0,1
%A _Paul Curtz_, Jan 26 2008
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