A077079 -id:A077079 - cp's OEIS frontend

A077078 Number of different absolute values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Dipole moments of n positive and n negative equally spaced charges on a circle.

1, 2, 3, 6, 11, 20, 53, 130, 196, 725, 1990, 2437, 14466, 38697, 18878

Offset: 1

Wouter Meeussen, Oct 27 2002

Starting from n ones and n zero's cf. A006840, the simplification can remove subsets of ones with vector sum equal to zero, thus lowering the count of ones in the simplified bracelet.

The count is lower than A077079 because the absolute vector sums can be equal without being symmetrically related.

			The 35 bracelets for n=6 simplify to the following 20: {0,0,0,0,0,0,0,0,0,0,0,0}, {0,0,0,0,0,0,0,0,0,0,0,1}, {0,0,0,0,0,0,0,0,0,0,1,1}, {0,0,0,0,0,0,0,0,0,1,0,1}, {0,0,0,0,0,0,0,0,0,1,1,1}, {0,0,0,0,0,0,0,0,1,0,0,1}, {0,0,0,0,0,0,0,0,1,0,1,1}, {0,0,0,0,0,0,0,0,1,1,1,1}, {0,0,0,0,0,0,0,1,0,1,0,1}, {0,0,0,0,0,0,0,1,0,1,1,1}, {0,0,0,0,0,0,1,0,0,0,0,1}, {0,0,0,0,0,0,1,0,0,1,0,1}, {0,0,0,0,0,0,1,0,1,0,1,1}, {0,0,0,0,0,0,1,0,1,1,0,1}, {0,0,0,0,0,0,1,1,1,1,1,1}, {0,0,0,0,1,0,0,0,0,1,0,1}, {0,0,0,0,1,0,0,1,0,1,0,1}, {0,0,0,0,1,0,1,0,0,1,0,1}, {0,0,0,0,1,0,1,1,1,1,0,1}, {0,0,1,0,1,0,1,1,0,1,0,1}

Cf. A077079, A006840.

A045611 Number of different energy states of n positive and n negative charges on a necklace. Different sets of distances between n points chosen from 2n equally spaced points on a circle.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 35, 85, 254, 701, 2377, 7944, 25220, 95910, 332300, 1164825, 4379920, 16649851, 58473414

Offset: 0

Wouter Meeussen

			a(10) = 2377 because the multiplicities {1; 2; 5; 10; 20; 40; 60; 80; 120} occur {1; 1; 3; 22; 362; 1855; 1; 130; 2} times and 2377 = 1 + 1 + 3 + 22 + 362 + 1855 + 1 + 130 + 2. Total number of configurations = 1*1 + 1*2 + 3*5 + 22*10 + .. = 92378 (see C(2n + 1, n + 1)=A001700).

Sean A. Irvine, Java program (github)
Index entries for sequences related to necklaces

For n=0..7 this is equal to A006840. Cf. A045610, A045611, A045612, A006840, A077078, A077079.

n=4; perm=Permutations[ Table[ -1, {n-1} ]~Join~Table[ 1, {n} ] ]; res=Table[ (Prepend[ #, -1 ].RotateRight[ Prepend[ #, -1 ], k ])/2, {k, 1, n} ]&/@perm; kort=Length[ Length/@Split[ Sort[ res ] ] ]

a(15) corrected and a(16)-a(18) from Sean A. Irvine, Mar 17 2021

A103441 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 set of distances among the white beads are different.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 3, 3, 1, 1, 3, 4, 4, 3, 1, 1, 4, 5, 7, 5, 4, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 5, 8, 16, 13, 16, 8, 5, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 6, 12, 28, 35, 35, 35, 28, 12, 6, 1, 1, 6, 14, 34, 57, 74, 74, 57, 34, 14, 6, 1, 1, 7, 16, 47, 73, 120, 85, 120, 73

Offset: 2

Wouter Meeussen, Feb 06 2005

If two bracelets can be made to coincide by rotation or flipping over they necessarily have the same set of distances, but the reverse is obviously not true.
Offset is 2, since exactly two colors are required, ergo at least two beads.
T[2n,n] equals A045611. Row sums equal A103442.
Same as A052307, except for bracelets such as {0,0,0,1,1,0,1,1} and{0,0,1,0,0,1,1,1}, that both have the same set of distances between the "1" beads: 4 d[0]+ 4 d[1]+ 2 d[2]+ 4 d[3]+ 2 d[4], where d[k] represents the unidirectional distance between two beads k places apart.

			Table starts as
  1;
  1,1;
  1,2,1;
  1,2,2,1;
  ...

Cf. A052307, A045611, A077078, A077079, A103442.

Needs[DiscreteMath`NewCombinatorica`]; f[bi_]:=DeleteCases[bi Range[Length[bi]], 0]; dist[li_, l_]:=Plus@@Flatten[Outer[d[Min[ #, l-# ]&@Mod[Abs[ #1-#2], l, 0]]&, li, li]]; Table[Length[Union[(dist[f[ #1], n]&)/@ListNecklaces[n, Join[1+0*Range[i], 0*Range[n-i]], Dihedral]]], {n, 2, 16}, {i, 1, n-1}]

cp's OEIS Frontend

A077078 Number of different absolute values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Dipole moments of n positive and n negative equally spaced charges on a circle.

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A045611 Number of different energy states of n positive and n negative charges on a necklace. Different sets of distances between n points chosen from 2n equally spaced points on a circle.

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A103441 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 set of distances among the white beads are different.

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Mathematica