A077078 -id:A077078 - cp's OEIS frontend

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.

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

Offset: 0

			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

A077079 Number of inequivalent bracelets from A006840 with the additional equivalence condition that subsets of 1-beads whose position vectors add to zero can be removed. Different values of vector sums of (-1)^(k/n) with k taking n values in 1..2n up to rotation and reflection.

Original entry on oeis.org

1, 2, 3, 6, 11, 20, 53, 130, 199, 784, 2135, 2649, 15695, 43085, 32764

Offset: 1

Wouter Meeussen, Oct 27 2002

At n=15 the sequence decreases because of the large number of divisors of 30.

Identical to A077078 up to n=9. Cf. A006840.

lowest[li_] := First[Sort[Join[NestList[RotateRight, li, 2n-1], NestList[RotateRight, 1-li, 2n-1], NestList[RotateRight, Reverse@li, 2n-1], NestList[RotateRight, 1-Reverse@li, 2n-1]]]]; ker[n_, k_] := Flatten[Table[Join[{1}, 0Range[ -1+2n/k]], {k}]]; ingekort[li_] := Module[{temp, divi}, len=Length[li]; temp=li-(liRotateRight[li, len/2]); divi=First/@FactorInteger[len]; Table[d=divi[[s]]; k=ker[len/2, d]; temp=Fold[kort[ #1, #2]&, temp, NestList[RotateRight, k, len/d-1]], {s, Length[divi], 2, -1}]; lowest[temp]]; kort[q_, k_] := If[(q.k>=Floor[d/2+1])&&(q.RotateRight[k-kq, len/2]===0), q-kq+RotateRight[k-kq, len/2], q]; Length[inequiv=Union[ingekort/@ListOfBraceletsA006840]]

A103692 Row sums of A103691.

Original entry on oeis.org

1, 2, 4, 6, 11, 16, 24, 44, 59, 124, 122, 372, 357, 966, 898, 3926, 1634, 13660, 6207, 32656

Offset: 2

Wouter Meeussen, Feb 12 2005

Cf. A077078; A005648; A052307, A103692.

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) f[bi_] := DeleteCases[ bi*Range[ Length[bi]], 0]; vec[li_, l_]:= Abs[Plus @@ N[Exp[2*Pi*I*f[li]/l], 24]]; Plus @@@ 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, 17}, {i, n-1}]

a(17)-a(21) from Robert G. Wilson v, Feb 14 2005

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}]

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.

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, 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

Wouter Meeussen, Feb 12 2005

Offset is 2, since exactly two colors are required, ergo at least two beads.

T[2n,n] equals A077078. Row sums equal A103692.

			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;
  ...

Cf. A077078; A005648; A052307, A103692.

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}]

A077013 Number of different argument values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Arguments are considered different only up to rotation and reflection.

Original entry on oeis.org

1, 2, 1, 2, 3, 7, 14, 43, 65, 292, 992, 1154

Offset: 1

Wouter Meeussen, Nov 28 2002

Arg[0] is taken as being equal to 0.

			The arguments up to rotation and reflection for n=6 are {0, 0.17013, 0.206867, 0.293133, 0.32987, 0.385502, 0.5} or exactly {0, 1/2, 5/2-(6*ArcTan[2])/Pi, 5/2-(6*ArcTan[1+2/Sqrt[3]])/Pi, -3/2-(6*ArcTan[4-3*Sqrt[3]])/Pi, 3/2-(6*ArcTan[6- 3*Sqrt[3]])/Pi, -5/2+(6*ArcTan[4+3*Sqrt[3]])/Pi }

Cf. A077078, A006840.

cp's OEIS Frontend

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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A077079 Number of inequivalent bracelets from A006840 with the additional equivalence condition that subsets of 1-beads whose position vectors add to zero can be removed. Different values of vector sums of (-1)^(k/n) with k taking n values in 1..2n up to rotation and reflection.

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A103692 Row sums of A103691.

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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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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.

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A077013 Number of different argument values of vector sums generated by n vectors chosen from 2n (equally spaced) points around the unit circle. Arguments are considered different only up to rotation and reflection.

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