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A209485 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and avoiding the patterns z z+1 z+2 and z z-1 z-2.

%I A209485 #10 Mar 13 2017 04:25:32
%S A209485 1,1,2,1,3,1,1,4,4,4,1,5,7,15,4,1,6,12,35,38,11,1,7,17,72,140,136,15,
%T A209485 1,8,24,128,390,731,458,43,1,9,31,205,866,2606,3740,1781,77,1,10,40,
%U A209485 311,1702,7179,17771,20888,6912,199,1,11,49,448,3014,16660,60778,128598,118137
%N A209485 T(n,k) is the number of n-bead necklaces labeled with numbers -k..k allowing reversal, with sum zero and avoiding the patterns z z+1 z+2 and z z-1 z-2.
%C A209485 Table starts
%C A209485 ..1....1.....1......1......1.......1.......1........1........1........1.......1
%C A209485 ..2....3.....4......5......6.......7.......8........9.......10.......11......12
%C A209485 ..1....4.....7.....12.....17......24......31.......40.......49.......60......71
%C A209485 ..4...15....35.....72....128.....205.....311......448......618......829....1083
%C A209485 ..4...38...140....390....866....1702....3014.....4984.....7774....11620...16716
%C A209485 .11..136...731...2606...7179...16660...34233....64220...112263...185506..292759
%C A209485 .15..458..3740..17771..60778..168453..401634...857433..1679810..3074315.5321674
%C A209485 .43.1781.20888.128598.541494.1778878.4907310.11891268.26069478.52776268
%H A209485 R. H. Hardin, <a href="/A209485/b209485.txt">Table of n, a(n) for n = 1..148</a>
%F A209485 Empirical for row n:
%F A209485 n=2: a(k) = 2*a(k-1) - a(k-2).
%F A209485 n=3: a(k) = 2*a(k-1) - 2*a(k-3) + a(k-4).
%F A209485 n=4: a(k) = 3*a(k-1) - 3*a(k-2) + 2*a(k-3) - 3*a(k-4) + 3*a(k-5) - a(k-6).
%F A209485 n=5: a(k) = 2*a(k-1) - a(k-3) - 2*a(k-5) + 2*a(k-6) + a(k-8) - 2*a(k-10) + a(k-11).
%F A209485 n=6: a(k) = 5*a(k-1) - 10*a(k-2) + 11*a(k-3) - 10*a(k-4) + 11*a(k-5) - 10*a(k-6) + 5*a(k-7) - a(k-8) for k > 9.
%e A209485 Some solutions for n=6, k=6:
%e A209485 .-5...-4...-5...-6...-6...-5...-6...-4...-3...-6...-6...-3...-5...-5...-6...-4
%e A209485 ..0....0...-2...-3...-2...-4...-5...-3...-1....5....2...-2....0....2...-2....2
%e A209485 .-2...-2....2....4....0...-1....4....0...-1...-5....0...-2...-3...-4....6...-4
%e A209485 ..2....2...-2....3....1....5....3....4....1....0...-4....5....2....1...-4....4
%e A209485 ..5....0....5...-2....1....0....4....3...-2....0....6....0....5....0....0...-4
%e A209485 ..0....4....2....4....6....5....0....0....6....6....2....2....1....6....6....6
%Y A209485 Row 3 is A074148.
%Y A209485 Row 4 is A209345.
%K A209485 nonn,tabl
%O A209485 1,3
%A A209485 _R. H. Hardin_, Mar 09 2012