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A204453 Period length 14: [0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1] repeated.

%I A204453 #27 Sep 11 2018 03:20:03
%S A204453 0,1,2,3,4,5,6,0,6,5,4,3,2,1,0,1,2,3,4,5,6,0,6,5,4,3,2,1,0,1,2,3,4,5,
%T A204453 6,0,6,5,4,3,2,1,0,1,2,3,4,5,6,0,6,5,4,3,2,1,0,1,2,3,4,5,6,0,6,5,4,3,
%U A204453 2,1,0,1,2,3,4,5,6,0,6,5,4,3,2,1,0,1,2,3,4,5
%N A204453 Period length 14: [0, 1, 2, 3, 4, 5, 6, 0, 6, 5, 4, 3, 2, 1] repeated.
%C A204453 This sequence can be continued periodically for negative values of n, and then a(-n) = a(n).
%C A204453 This is the seventh sequence of a k-family of sequences P_k, k>=1, which starts with A000007(n+1), n>=0, (the 0-sequence), A000035, A193680, A193682, A203572, A for k=1..6, respectively.
%C A204453 See a comment on A203571 for the general case of the P_k sequences. For a(n)=P_7(n) the nonnegative members of the equivalence classes [0], [1],...,[6], defined by p==q iff P_7(p)=P_7(q), are found in the array A113807 if there the last class [7], starting with 7, is replaced by 0,7,14,..., to become the first class [0] (nonnegative part).
%H A204453 <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1).
%F A204453 a(n) = n(mod 7) if (-1)^floor(n/7)=+1 else (7-n)(mod 7), n>=0. (-1)^floor(n/7) is the sign corresponding to the parity of the quotient floor(n/7). This quotient is sometimes denoted by n\7.
%F A204453 O.g.f.: x*(1+2*x+3*x^2+4*x^3+5*x^4+6*x^5+6*x^7+5*x^8+4*x^9+ 3*x^10+2*x^11+x^12)/(1-x^14).
%F A204453 a(n) = (7*m*(m^4-21*m^3+175*m^2-735*m+1624)*((-1)^floor(n/7)-1)-10908*(-1)^floor(n/7)+12348)*m/1440 where m = n-7*floor(n/7). - _Luce ETIENNE_, Oct 13 2017
%e A204453 a(16) = 16(mod 7) = 2 because 16\7 = floor(16/7)=2 is even; the sign is +1.
%e A204453 a(9) = (7-9)(mod 7) = 5 because 9\7 = floor(9/7)=1 is odd; the sign is -1.
%Y A204453 Cf. A203572 (k=6), A113807, A010876.
%K A204453 nonn,easy
%O A204453 0,3
%A A204453 _Wolfdieter Lang_, Jan 17 2012