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%I #11 Jul 09 2018 08:23:04
%S 5,40,145,400,883,1724,3045,5026,7827,11684,16795,23446,31879,42430,
%T 55379,71118,89965,112362,138671,169384,204901,245770,292429,345476,
%U 405393,472828,548301,632516,726031,829600,943825,1069510,1207295
%N Number of 5-bead necklaces labeled with numbers -n..n allowing reversal, with sum zero with no three beads in a row equal.
%C Row 5 of A209344.
%H R. H. Hardin, <a href="/A209346/b209346.txt">Table of n, a(n) for n = 1..210</a>
%F Empirical: a(n) = 2*a(n-1) + a(n-2) - 3*a(n-3) - a(n-4) + a(n-5) + 3*a(n-6) - a(n-7) - 2*a(n-8) + a(n-9).
%F Empirical g.f.: x*(5 + 30*x + 60*x^2 + 85*x^3 + 63*x^4 + 28*x^5 + 4*x^6 + x^7) / ((1 - x)^5*(1 + x)^2*(1 + x + x^2)). - _Colin Barker_, Jul 09 2018
%e Some solutions for n=10:
%e -9 -7 -10 -5 -10 -10 -8 -7 -8 -7 -9 -7 -4 -6 -10 -8
%e 5 4 -4 -1 -4 -5 -7 -3 1 0 -4 2 -2 1 -4 -3
%e 7 -3 -5 3 4 -1 8 -3 6 -1 9 -3 -2 4 10 8
%e -9 -1 10 3 5 10 3 3 -1 3 3 2 10 -2 6 1
%e 6 7 9 0 5 6 4 10 2 5 1 6 -2 3 -2 2
%Y Cf. A209344.
%K nonn
%O 1,1
%A _R. H. Hardin_, Mar 06 2012