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%I #13 Jul 07 2018 19:30:05
%S 2,11,34,88,187,358,625,1023,1584,2355,3374,4700,6377,8476,11049,
%T 14175,17916,22361,27580,33672,40713,48816,58063,68577,80448,93809,
%U 108760,125442,143963,164476,187095,211985,239266,269115,301660,337086,375531
%N Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.
%C Row 5 of A209032.
%H R. H. Hardin, <a href="/A209033/b209033.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - a(n-5) - 2*a(n-6) + 2*a(n-7) + a(n-8) - 2*a(n-9) + 2*a(n-10) - 2*a(n-12) + a(n-13).
%F Empirical g.f.: x*(2 + 7*x + 12*x^2 + 24*x^3 + 29*x^4 + 32*x^5 + 32*x^6 + 23*x^7 + 12*x^8 + 9*x^9 - x^11 + x^12) / ((1 - x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2 + x^3 + x^4)). - _Colin Barker_, Jul 07 2018
%e Some solutions for n=6;
%e -2 -5 -2 -5 -3 -4 -6 -4 -3 -1 -6 -3 -2 -4 -1 -2
%e 0 -1 0 -1 0 0 -1 -1 0 -1 -1 -1 -2 0 0 -2
%e -1 0 -2 2 0 3 5 3 -3 -1 4 0 3 -1 0 -1
%e -1 5 0 4 0 0 2 1 3 -1 4 4 -2 3 0 3
%e 4 1 4 0 3 1 0 1 3 4 -1 0 3 2 1 2
%Y Cf. A209032.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 04 2012
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