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A209033 Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero and first differences in -n..n.

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