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%I #13 Jul 07 2018 08:51:21
%S 1,3,4,8,11,18,24,35,45,61,76,98,119,148,176,213,249,295,340,396,451,
%T 518,584,663,741,833,924,1030,1135,1256,1376,1513,1649,1803,1956,2128,
%U 2299,2490,2680,2891,3101,3333,3564,3818,4071,4348,4624,4925,5225,5551,5876
%N Number of 4-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first and second differences in -n..n.
%C Row 4 of A208970.
%H R. H. Hardin, <a href="/A208971/b208971.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6).
%F Conjectures from _Colin Barker_, Jul 07 2018: (Start)
%F G.f.: x*(1 + x - 3*x^2 + x^3 + 2*x^4 - x^5) / ((1 - x)^4*(1 + x)^2).
%F a(n) = (2*n^3 + 6*n^2 + 28*n + 48) / 48 for n even.
%F a(n) = (2*n^3 + 6*n^2 + 22*n + 18) / 48 for n odd.
%F (End)
%e All solutions for n=5:
%e -2 -1 -2 -2 -1 -1 -1 0 -2 -2 -1
%e 0 -1 0 -1 0 0 1 0 -1 -2 -1
%e 2 0 1 1 0 1 -1 0 2 2 1
%e 0 2 1 2 1 0 1 0 1 2 1
%Y Cf. A208970.
%K nonn
%O 1,2
%A _R. H. Hardin_, Mar 03 2012