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A260686 Period 6 zigzag sequence, repeat [0, 1, 2, 3, 2, 1].

%I A260686 #55 May 25 2024 10:45:22
%S A260686 0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,
%T A260686 2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,
%U A260686 2,3,2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1
%N A260686 Period 6 zigzag sequence, repeat [0, 1, 2, 3, 2, 1].
%C A260686 Decimal expansion of 37/3003. - _Elmo R. Oliveira_, Mar 06 2024
%H A260686 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1).
%F A260686 G.f.: x*(1 + x + x^2) / (1 - x + x^3 - x^4).
%F A260686 a(n) = a(n-1) - a(n-3) + a(n-4) for n > 3.
%F A260686 a(n) = Sum_{i=1..n} (-1)^floor((i-1)/3) for n > 0.
%F A260686 a(n+1) = a(n) + A130151(n).
%F A260686 a(2n) = 2*A011655(n), a(2n+1) = A109007(n+2).
%F A260686 a(n) = 1 + (1 - (-1)^n)/2 - (-1)^floor((n+1)/3). - _Bruno Berselli_, Nov 16 2015
%F A260686 a(n) = sin(n*Pi/6)^2*(11+4*cos(n*Pi/3)+2*cos(2*n*Pi/3))/3. - _Wesley Ivan Hurt_, Jun 17 2016
%F A260686 a(n) = a(n-6) for n >= 6. - _Wesley Ivan Hurt_, Sep 07 2022
%F A260686 a(n) = sqrt(n^2 mod 12) = sqrt(A070435(n)). - _Nicolas Bělohoubek_, May 24 2024
%p A260686 A260686:=n->[0, 1, 2, 3, 2, 1][(n mod 6)+1]: seq(A260686(n), n=0..100);
%t A260686 CoefficientList[Series[(x + x^2 + x^3)/(1 - x + x^3 - x^4), {x, 0, 100}], x]
%t A260686 Table[1 + (1 - (-1)^n)/2 - (-1)^Floor[(n + 1)/3], {n, 0, 100}] (* _Bruno Berselli_, Nov 16 2015 *)
%t A260686 PadRight[{}, 120, {0, 1, 2, 3, 2, 1}] (* _Vincenzo Librandi_, Nov 17 2015 *)
%o A260686 (PARI) concat(0, Vec((x+x^2+x^3)/(1-x+x^3-x^4) + O(x^100))) \\ _Altug Alkan_, Nov 15 2015
%o A260686 (Magma) [1+(1-(-1)^n)/2-(-1)^Floor((n+1)/3): n in [0..100]]; // _Bruno Berselli_, Nov 16 2015
%o A260686 (Magma) &cat[[0,1,2,3,2,1]: n in [0..15]]; // _Vincenzo Librandi_, Nov 17 2015
%Y A260686 Period k zigzag sequences: A000035 (k=2), A007877 (k=4), this sequence (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
%Y A260686 Cf. A011655, A070435, A109007, A130151.
%K A260686 nonn,easy
%O A260686 0,3
%A A260686 _Wesley Ivan Hurt_, Nov 15 2015