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Number of 5-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.
2

%I #12 Mar 18 2018 15:43:39

%S 11,77,291,791,1761,3431,6077,10021,15631,23321,33551,46827,63701,

%T 84771,110681,142121,179827,224581,277211,338591,409641,491327,584661,

%U 690701,810551,945361,1096327,1264691,1451741,1658811,1887281,2138577,2414171

%N Number of 5-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero.

%C Row 5 of A208597.

%H R. H. Hardin, <a href="/A208599/b208599.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (23/12)*n^4 + (23/6)*n^3 + (37/12)*n^2 + (7/6)*n + 1.

%F Conjectures from _Colin Barker_, Mar 07 2018: (Start)

%F G.f.: x*(11 + 22*x + 16*x^2 - 4*x^3 + x^4) / (1 - x)^5.

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

%F (End)

%e Some solutions for n=4:

%e -4 -3 -4 -4 -3 -3 -3 -3 -3 -4 -4 -4 -4 -4 -4 -1

%e -1 2 3 2 0 2 1 -1 4 2 0 -1 4 0 0 0

%e 4 -2 2 -4 -3 2 2 2 0 0 0 4 -3 2 -1 1

%e -2 3 0 3 3 -1 -2 0 1 -2 2 1 4 1 2 -1

%e 3 0 -1 3 3 0 2 2 -2 4 2 0 -1 1 3 1

%Y Cf. A208597.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 29 2012