A103691 Triangle read by rows: T(n,k) = number of bracelets of n beads (necklaces that can be flipped over) with exactly two colors and k white beads, for which the length (or abs value) of sum of the position vectors of the white beads are different.
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 4, 6, 4, 4, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 5, 7, 11, 11, 11, 7, 5, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 6, 10, 16, 18, 20, 18, 16, 10, 6, 1, 1, 6, 14, 34, 57, 74, 74, 57, 34, 14, 6, 1, 1, 7, 14, 33, 44, 53, 53, 53, 44, 33
Offset: 2
Examples
T[8,3]=4 because of the 5 bracelets {1,1,1,0,0,0,0,0}, {0,0,0,0,1,0,1,1}, {0,0,0,1,0,0,1,1},{0,0,0,1,0,1,0,1} and {0,0,1,0,0,1,0,1}, the third and the fourth have equal absolute vector sums, length 1. Table starts as: 1; 1,1; 1,2,1; 1,2,2,1; ...
Programs
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Mathematica
Needs[DiscreteMath`NewCombinatorica`]; f[bi_]:=DeleteCases[bi*Range[Length[bi]], 0]; vec[li_, l_]:= Abs[Plus@@ N[Exp[2*Pi*I*f[li]/l], 24]]; Table[Length[Union[(vec[ #, n]&)/@ ListNecklaces[n, Join[1+0*Range[i], 0*Range[n-i]], Dihedral], SameTest->(Abs[ #1-#2]<10^-18&)]], {n, 2, 16}, {i, 1, n-1}]
Comments