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%I #18 Mar 19 2018 04:09:45
%S 1,1,3,5,7,11,15,19,25,31,37,45,53,61,71,81,91,103,115,127,141,155,
%T 169,185,201,217,235,253,271,291,311,331,353,375,397,421,445,469,495,
%U 521,547,575,603,631,661,691,721,753,785,817,851,885,919,955,991,1027,1065,1103,1141
%N Number of 3-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first differences in -n..n.
%H Alois P. Heinz, <a href="/A208994/b208994.txt">Table of n, a(n) for n = 0..10000</a> (first 210 terms from R. H. Hardin)
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1)
%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5).
%F From _Alois P. Heinz_, Mar 07 2018: (Start)
%F a(n) = 1 + floor(n*(n+1)/3).
%F G.f.: -(x^2+1)*(x^2-x+1)/((x^2+x+1)*(x-1)^3). (End)
%e All 15 solutions for n=6:
%e -2 -1 -3 -2 -2 -1 -1 -3 -2 -3 -4 0 -2 -3 -2
%e 1 -1 3 0 3 1 0 2 -1 0 2 0 -2 1 2
%e 1 2 0 2 -1 0 1 1 3 3 2 0 4 2 0
%Y Row n=3 of A208993.
%K nonn
%O 0,3
%A _R. H. Hardin_, Mar 04 2012
%E a(0)=1 prepended by _Alois P. Heinz_, Mar 07 2018