%I #31 Mar 03 2024 09:51:30
%S 0,1,2,3,4,5,6,5,4,3,2,1,0,1,2,3,4,5,6,5,4,3,2,1,0,1,2,3,4,5,6,5,4,3,
%T 2,1,0,1,2,3,4,5,6,5,4,3,2,1,0,1,2,3,4,5,6,5,4,3,2,1,0,1,2,3,4,5,6,5,
%U 4,3,2,1,0,1,2,3,4,5,6,5,4,3,2,1,0,1
%N Period 12 zigzag sequence: repeat [0,1,2,3,4,5,6,5,4,3,2,1].
%C a(n)/36 is the probability that the sum shown after rolling a pair of standard dice is 1+(n mod 12). - _Mathew Englander_, Jul 11 2022
%C Decimal expansion of 37037/3000003. - _Elmo R. Oliveira_, Mar 03 2024
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,-1,1).
%F G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7).
%F a(n) = a(n-1) - a(n-6) + a(n-7) for n>6.
%F a(n) = abs(n - 12*round(n/12)).
%F a(n) = Sum_{i=1..n} (-1)^floor((i-1)/6).
%F a(2n) = a(10n) = 2*A260686(n), a(2n+1) = A110551(n).
%F a(3n) = 3*A007877(n), a(4n) = a(8n) = 4*A011655(n).
%F a(6n) = A010677(n) = 6*A000035(n).
%F a(n) = a(n-12) for n >= 12. - _Wesley Ivan Hurt_, Sep 07 2022
%p A271832:=n->[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1][(n mod 12)+1]: seq(A271832(n), n=0..300);
%t CoefficientList[Series[x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7), {x, 0, 100}], x]
%o (Magma) &cat[[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]: n in [0..10]];
%o (PARI) lista(nn) = for(n=0, nn, print1(abs(n-12*round(n/12)), ", ")); \\ _Altug Alkan_, Apr 15 2016
%Y Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), this sequence (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).
%Y Cf. A010677, A011655, A110551.
%K nonn,easy
%O 0,3
%A _Wesley Ivan Hurt_, Apr 15 2016