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%I #13 Jul 07 2018 06:15:43
%S 18,167,828,2821,7582,17339,35288,65769,114442,188463,296660,449709,
%T 660310,943363,1316144,1798481,2412930,3184951,4143084,5319125,
%U 6748302,8469451,10525192,12962105,15830906,19186623,23088772,27601533,32793926
%N Number of 6-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero.
%C Row 6 of A208825.
%H R. H. Hardin, <a href="/A208827/b208827.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = (22/15)*n^5 + (11/3)*n^4 + (14/3)*n^3 + (13/3)*n^2 + (43/15)*n + 1.
%F Conjectures from _Colin Barker_, Jul 07 2018: (Start)
%F G.f.: x*(18 + 59*x + 96*x^2 - 2*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=3:
%e -3 -3 -3 -2 -2 -3 -3 -2 -3 -3 -3 -3 -2 -3 -3 -2
%e -1 0 -2 0 -1 -2 -1 -1 0 -3 -1 -1 -1 2 -2 0
%e 3 1 0 0 3 2 0 1 -2 1 0 2 -2 -2 3 1
%e -2 -1 3 0 0 3 2 -2 -1 3 0 -1 2 -2 2 -2
%e 3 0 3 0 0 -3 -1 2 3 0 3 3 0 2 -1 0
%e 0 3 -1 2 0 3 3 2 3 2 1 0 3 3 1 3
%Y Cf. A208825.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 01 2012
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