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%I #13 Mar 18 2018 17:54:04
%S 26,297,1564,5457,14838,34153,69784,130401,227314,374825,590580,
%T 895921,1316238,1881321,2625712,3589057,4816458,6358825,8273228,
%U 10623249,13479334,16919145,21027912,25898785,31633186,38341161,46141732,55163249
%N Number of 6-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.
%C Row 6 of A208597.
%H R. H. Hardin, <a href="/A208600/b208600.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (44/15)*n^5 + (22/3)*n^4 + (23/3)*n^3 + (14/3)*n^2 + (12/5)*n + 1.
%F Conjectures from _Colin Barker_, Mar 07 2018: (Start)
%F G.f.: x*(26 + 141*x + 172*x^2 + 8*x^3 + 6*x^4 - x^5) / (1 - x)^6.
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
%F (End)
%e Some solutions for n=4:
%e -4 -4 -4 -3 -4 -2 -4 -4 -4 -4 -3 -4 -4 -3 -3 -1
%e 4 3 2 2 -3 -2 -1 2 0 -1 3 4 2 -1 3 0
%e 0 2 -1 3 -2 2 0 -3 4 2 -3 -2 1 2 0 -1
%e 1 1 -3 0 3 0 -1 0 -1 1 3 0 3 -2 -2 0
%e -1 1 3 0 4 -2 4 3 4 4 -3 1 -2 4 0 -1
%e 0 -3 3 -2 2 4 2 2 -3 -2 3 1 0 0 2 3
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 29 2012