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%I #16 Jul 07 2018 19:31:24
%S 2,7,16,33,58,95,144,209,290,391,512,657,826,1023,1248,1505,1794,2119,
%T 2480,2881,3322,3807,4336,4913,5538,6215,6944,7729,8570,9471,10432,
%U 11457,12546,13703,14928,16225,17594,19039,20560,22161,23842,25607,27456,29393
%N Number of 4-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first differences in -n..n.
%C Row 4 of A208993.
%H R. H. Hardin, <a href="/A208995/b208995.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*(2 + x - x^2 + 3*x^3 - x^4) / ((1 - x)^4*(1 + x)).
%F a(n) = (12 + 8*n + 6*n^2 + 4*n^3) / 12 for n even.
%F a(n) = (6 + 8*n + 6*n^2 + 4*n^3) / 12 for n odd.
%F (End)
%e Some solutions for n=6:
%e -3 -3 -4 -3 -2 -2 -4 -2 -1 -3 -2 -2 -4 -2 -3 -2
%e 0 -1 -1 1 -2 0 0 2 -1 1 4 1 -1 0 3 0
%e 2 2 4 3 4 0 5 0 1 2 -1 2 3 -1 2 2
%e 1 2 1 -1 0 2 -1 0 1 0 -1 -1 2 3 -2 0
%Y Cf. A208993.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 04 2012
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