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T(n,k) = number of n-bead necklaces labeled with numbers -k..k not allowing reversal, with sum zero and avoiding the pattern z z+1 z+2.
12

%I #11 Mar 19 2018 04:10:05

%S 1,1,2,1,3,2,1,4,6,5,1,5,12,22,8,1,6,20,57,70,19,1,7,30,122,276,265,

%T 39,1,8,42,223,766,1465,1007,95,1,9,56,366,1722,5239,8008,4055,223,1,

%U 10,72,563,3378,14431,37108,45408,16593,554,1,11,90,820,6006,33469,125428

%N T(n,k) = number of n-bead necklaces labeled with numbers -k..k not allowing reversal, with sum zero and avoiding the pattern z z+1 z+2.

%C Table starts

%C 1 1 1 1 1 1 1 1 1 1

%C 2 3 4 5 6 7 8 9 10 11

%C 2 6 12 20 30 42 56 72 90 110

%C 5 22 57 122 223 366 563 820 1143 1544

%C 8 70 276 766 1722 3378 6006 9928 15514 23180

%C 19 265 1465 5239 14431 33469 68723 128845 225127 371859

%C 39 1007 8008 37108 125428 344618 817594 1738838 3397900 6206228

%C 95 4055 45408 271732 1125484 3660116 10030238 24194286 52868486 106770774

%H R. H. Hardin, <a href="/A209073/b209073.txt">Table of n, a(n) for n = 1..166</a>

%e Some solutions for n=6, k=6:

%e -6 -5 -5 -4 -6 -6 -4 -5 -6 -5 -5 -6 -6 -4 -4 -6

%e 5 6 -1 2 -2 1 0 2 -1 4 -1 3 -1 -3 5 4

%e 2 1 0 1 -5 5 4 -1 2 0 -5 3 5 5 -2 -4

%e -3 -4 5 1 4 -4 3 -5 4 2 2 0 2 1 2 3

%e 3 3 -1 -2 6 0 0 5 -2 1 4 -6 5 2 0 -1

%e -1 -1 2 2 3 4 -3 4 3 -2 5 6 -5 -1 -1 4

%Y Row 3 is A002378.

%Y Row 4 is A208946.

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Mar 04 2012