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%I #16 Jul 07 2018 19:31:37
%S 1,3,5,10,16,26,38,55,75,101,131,168,210,260,316,381,453,535,625,726,
%T 836,958,1090,1235,1391,1561,1743,1940,2150,2376,2616,2873,3145,3435,
%U 3741,4066,4408,4770,5150,5551,5971,6413,6875,7360,7866,8396,8948,9525,10125
%N Number of 4-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first and second differences in -n..n.
%C Row 4 of A209007.
%H R. H. Hardin, <a href="/A209008/b209008.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
%F Conjectures from _Colin Barker_, Jul 07 2018: (Start)
%F G.f.: x*(1 - 2*x^2 + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).
%F a(n) = (4*n^3 + 6*n^2 + 20*n + 48) / 48 for n even.
%F a(n) = (4*n^3 + 6*n^2 + 20*n + 18) / 48 for n odd.
%F (End)
%e Some solutions for n=6:
%e -2 0 -3 -3 -1 -3 -2 -3 -2 -3 -1 -2 -1 -2 -1 -2
%e -2 0 0 -2 1 -3 1 -1 -1 1 -1 1 0 0 0 2
%e 2 0 3 3 -1 3 0 3 2 3 1 1 -1 0 0 0
%e 2 0 0 2 1 3 1 1 1 -1 1 0 2 2 1 0
%Y Cf. A209007.
%K nonn
%O 1,2
%A _R. H. Hardin_, Mar 04 2012
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