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A203398 T(n,k), a triangular array read by rows, is the number of classes of equivalent 2-color n-bead necklaces (turning over is not allowed) that have k necklaces. 1
2, 2, 1, 2, 0, 2, 2, 1, 0, 3, 2, 0, 0, 0, 6, 2, 1, 2, 0, 0, 9, 2, 0, 0, 0, 0, 0, 18, 2, 1, 0, 3, 0, 0, 0, 30, 2, 0, 2, 0, 0, 0, 0, 0, 56, 2, 1, 0, 0, 6, 0, 0, 0, 0, 99, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 186, 2, 1, 2, 3, 0, 9, 0, 0, 0, 0, 0, 335 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Equivalently, the cyclic group of order n acts on the set of length n binary sequences.  T(n,k) is the number of orbits that have k elements.

LINKS

Table of n, a(n) for n=1..78.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Frank Ruskey, Combinatorial Generation Algorithm Algorithm 4.24, p. 95.

EXAMPLE

  2

  2  1

  2  0  2

  2  1  0  3

  2  0  0  0  6

  2  1  2  0  0  9

  2  0  0  0  0  0  18

  2  1  0  3  0  0  0  30

  2  0  2  0  0  0  0  0  56

  2  1  0  0  6  0  0  0  0  99

  2  0  0  0  0  0  0  0  0  0  186

  2  1  2  3  0  9  0  0  0  0  0   335

MATHEMATICA

Needs["Combinatorica`"];

f[list_] := Sort[NestList[RotateLeft, list, Length[list]-1]]; Flatten[Table[Distribution[Map[Length, Map[Union, Union[Map[f, Strings[{0, 1}, n]]]]], Range[n]], {n, 1, 12}]]

CROSSREFS

A000031 (row sums), T(n,n) = A001037, T(n,n) = A064535 when n is prime, T(n,k) = A001037(k) when k divides n.

Cf. A203399.

Sequence in context: A104405 A156381 A089077 * A225064 A130071 A321373

Adjacent sequences:  A203395 A203396 A203397 * A203399 A203400 A203401

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Jan 01 2012

STATUS

approved

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Last modified November 21 14:18 EST 2019. Contains 329371 sequences. (Running on oeis4.)