This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A203399 #93 Aug 16 2025 21:08:46
%S A203399 2,0,2,1,0,0,2,0,2,0,0,0,2,1,0,3,0,0,0,0,2,0,0,0,6,0,0,0,0,0,2,1,2,0,
%T A203399 0,7,0,0,0,0,0,1,2,0,0,0,0,0,14,0,0,0,0,0,0,2,2,1,0,3,0,0,0,18,0,0,0,
%U A203399 0,0,0,0,6,2,0,2,0,0,0,0,0,28,0,0,0,0,0,0,0,0,14,2,1,0,0,6,0,0,0,0,39,0,0,0,0,0,0,0,0,0,30
%N A203399 T(n, k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is allowed) that contain k necklaces.
%C A203399 Equivalently, the dihedral group of order n acts on the set of length n binary sequences. T(n,k) is the number of orbits that contain k elements.
%C A203399 By "n-bead necklaces (turning over is allowed)" the author means "bracelets". By saying that the classes of these n-bead bracelets (turnover necklaces) "contain k necklaces" the author means that the classes "contain k marked necklaces". - _Petros Hadjicostas_, Jun 06 2019
%H A203399 Petros Hadjicostas, <a href="/A203399/b203399.txt">Table of n, a(n) for n = 1..210, flattened</a>.
%H A203399 Petros Hadjicostas, <a href="/A203399/a203399.pdf">Formulas for chiral bracelets</a>, 2019.
%H A203399 S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, <a href="http://www.socs.uoguelph.ca/~sawada/papers/fix-brace.pdf">Generating bracelets with fixed content</a>, 2011.
%H A203399 S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, <a href="https://doi.org/10.1016/j.tcs.2012.11.024">Generating bracelets with fixed content</a>, Theoretical Computer Science, Volume 475, 4 March 2013, Pages 103-112.
%H A203399 F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>
%H A203399 Frank Ruskey, <a href="https://web.archive.org/web/20171022155546/http://www.1stworks.com/ref/RuskeyCombGen.pdf">Combinatorial Generation Algorithm Algorithm 4.24, p. 95</a>.
%H A203399 Joe Sawada, <a href="https://doi.org/10.1137/S0097539700377037">Generating bracelets in constant amortized time</a>, SIAM J. Comput. 31 (2001), 259-268.
%H A203399 R. M. Thompson and R. T. Downs, <a href="https://www.geo.arizona.edu/xtal/group/pdf/acB57766.pdf">Systematic generation of all nonequivalent closest packed stacking sequences of length N using group theory</a>, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.
%F A203399 From _Petros Hadjicostas_, Jun 06 2019: (Start)
%F A203399 Conjectures: For n >= 1, let b(n) be the number of bracelets of two colors with n beads that are either periodic (period >= 2), or have reflection symmetry (achiral), or both. Then b(n) = A000029(n) - A032239(n) for n >= 3 with b(n) = A000029(n) for n = 1, 2. We have A000029(n) = 2^floor(-3 + n/2) * (7 - (-1)^n) + (1/(2*n)) * Sum_{d|n} phi(d) * 2^(n/d) for n >= 1 and A032239(n) = (1/2) * Sum_{d|n} mu(d) * (-2^floor(-2 + (n/(2*d))) * (7 - (-1)^(n/d)) + 2^(n/d)/n) for n >= 3.
%F A203399 For 1 <= k <= n, we conjecture that T(n, k) = Sum_{d|k} mu(d)*b(k/d) for k|n, and = 0 otherwise. Note that, if 3 <= n <= 11, we have T(n, k) = A056493(k) when k|n, but this is not true (for example) for n = 12 and n = 14. We have T(12, 12) = 82 <> 81 = A056493(12) and T(14, 14) = 177 <> 175 = A056493(14).
%F A203399 Apparently, T(n, 2*n) = A032239(n) for all n >= 3, and T(n, k) = 0 for n < k < 2*n. (End)
%F A203399 From _Petros Hadjicostas_, Jun 16 2019: (Start)
%F A203399 I ran the author's Mathematica program for n = 1..21 and I saw that the conjecture is OK except for n = 18 and n = 21. The program gives T(n=18, k=12) = 1 and T(n=18, k=18) = 742 while my conjecture implies that T(n=18, k=12) = 0 (since k = 12 does not divide n = 18) and T(n=18, k=18) = 743. In addition, the program gives T(n=21, k=14) = 2 and T(n=21, k=21) = 2030, while my conjecture implies that T(n=21, k=14) = 0 (since k = 14 does not divide n = 21) and T(n=21, k=21) = 2032. Apparently, my conjecture needs to be refined.
%F A203399 For n = 18, the single bracelet whose equivalence class has 12 marked necklaces is (BBWBWW)^3 (with period 3).
%F A203399 For n = 21, the two bracelets whose equivalence classes have 14 marked necklaces each are (BBWBWWW)^3 and (WWBWBBB)^3 (each with period 3). (End)
%e A203399 Triangle begins (with rows n >= 1 and columns k >= 1):
%e A203399 2 0
%e A203399 2 1 0 0
%e A203399 2 0 2 0 0 0
%e A203399 2 1 0 3 0 0 0 0
%e A203399 2 0 0 0 6 0 0 0 0 0
%e A203399 2 1 2 0 0 7 0 0 0 0 0 1
%e A203399 2 0 0 0 0 0 14 0 0 0 0 0 0 2
%e A203399 2 1 0 3 0 0 0 18 0 0 0 0 0 0 0 6
%e A203399 2 0 2 0 0 0 0 0 28 0 0 0 0 0 0 0 0 14
%e A203399 From _Petros Hadjicostas_, Jun 07 2019: (Start)
%e A203399 Consider all bracelets (turnover necklaces) of two colors (B and W) that can be generated using one of Ruskey's websites above. We keep his numbering, declare whether it has reflexive symmetry or not (achiral or chiral, resp.), and find its value of k (= number of different marked necklaces belonging to its equivalence class).
%e A203399 We have: (1) BBBBBB (k=1, achiral), (2) BBBBBW (k=6, achiral), (3) BBBBWW (k=6, achiral), (4) BBBWBW (k=6, achiral), (5) BBBWWW (k=6, achiral), (6) BBWBBW (k=3, achiral), (7) BBWBWW (k=12, chiral), (8) BBWWWW (k=6, achiral), (9) BWBWBW (k=2, achiral), (10) BWBWWWW (k=6, achiral), (11) BWWBWW (k=3, achiral), (12) BWWWWW (k=6, achiral), (13) WWWWWW (k=1, achiral).
%e A203399 Hence, only bracelet (7) has no reflection symmetry, and thus it is chiral. The k=12 marked necklaces of its equivalence class are as follows:
%e A203399 BBWBWW, WBBWBW, WWBBWB, BWWBBW, WBWWBB, BWBWWB, and their mirror images BWWBWB, BBWWBW, WBBWWB, BWBBWW, WBWBBW, WWBWBB.
%e A203399 We see that T(n=6, k=1) = 2, T(n=6, k=2) = 1, T(n=6, k=3) = 2, T(n=6, k=6) = 7, and T(n=6, k=12) = 1, which agree with line n=6 in the triangle above. (End)
%t A203399 Needs["Combinatorica`"];
%t A203399 f[list_]:= Sort[NestList[RotateLeft, list, Length[list]-1] ~Join~NestList[RotateLeft, Reverse[list], Length[list]-1]]; Flatten[Table[Distribution[Map[Length, Map[Union, Union[Map[f, Strings[{0, 1}, n]]]]], Range[2 n]], {n, 1, 10}]]
%Y A203399 Cf. A000029 (row sums), A032239 (T(n, 2n) for n >= 3), A056493, A203398.
%K A203399 nonn,tabf
%O A203399 1,1
%A A203399 _Geoffrey Critzer_, Jan 01 2012