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Number of 5-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first and second differences in -n..n.
1

%I #9 Mar 18 2018 17:45:17

%S 1,5,13,31,71,137,243,399,619,927,1329,1857,2525,3355,4385,5627,7121,

%T 8893,10971,13407,16213,19449,23143,27333,32083,37413,43389,50053,

%U 57447,65651,74683,84631,95537,107455,120475,134625,149997,166649,184637

%N Number of 5-bead necklaces labeled with numbers -n..n not allowing reversal, with sum zero and first and second differences in -n..n.

%C Row 5 of A209007.

%H R. H. Hardin, <a href="/A209009/b209009.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + a(n-5) - 2*a(n-6) + 2*a(n-8) - a(n-9) + a(n-10) - 2*a(n-11) + 2*a(n-13) - a(n-14) - a(n-15) + 2*a(n-16) - 2*a(n-18) + a(n-19).

%e Some solutions for n=6:

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

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

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

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

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

%K nonn

%O 1,2

%A _R. H. Hardin_, Mar 04 2012