A263658 Number of (0, 1)-necklaces with n zeros and n ones without zigzags (see reference for precise definition).

0, 0, 1, 1, 2, 3, 7, 12, 27, 57, 128, 285, 659, 1518, 3561, 8389, 19936, 47607, 114397, 276018, 669035, 1627491, 3973106, 9728991, 23892779, 58828866, 145201423, 359182693, 890350290, 2211257973, 5501701981, 13711368630, 34225162345, 85555609119, 214166692430, 536810116905

Offset

0,5

Comments

See page 16 in the reference.

A zigzag is a substring which is either 010 or 101. The necklaces 01 and 10 are considered to be with a zigzag. Necklaces do not allow turnover.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..100

E. Munarini and N. Z. Salvi, Circular Binary Strings without Zigzags, Integers: Electronic Journal of Combinatorial Number Theory 3 (2003), #A19.

Formula

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A263656(d) / 2. - Andrew Howroyd, Feb 26 2017

Example

For n=2 the necklace is 0011.
For n=3 the necklace is 000111.
For n=4 the necklaces are 00001111, 00110011.
For n=5 the necklaces are 0000011111, 0001110011, 0001100111.

Mathematica

(* b = A263656 *)
b[n_ /; n < 6] := {0, 0, 4, 6, 12, 30}[[n + 1]];
b[n_] := b[n] = (1/n)*(3*(n - 1)*b[n - 1] - 4*(n - 4)*b[n - 2] + (7*n - 27)*b[n - 3] - 6*b[n - 4] + (7*n - 37)*b[n - 5] - 3*(n - 6)*b[n - 6]);
a[0] = 0;
a[n_] := (1/n)*DivisorSum[n, EulerPhi[n/#] * b[#]/2&];
Array[a, 36, 0] (* Jean-François Alcover, Sep 11 2017, after Andrew Howroyd *)

Crossrefs

Main diagonal of A263657.

Cf. A007039, A263655, A263656, A263659.

Sequence in context: A099163 A000676 A283823 * A296517 A182692 A203837

Adjacent sequences: A263655 A263656 A263657 * A263659 A263660 A263661

Keyword

nonn

Author

Felix Fröhlich, Oct 23 2015

Extensions

Offset corrected and a(21)-a(35) from Andrew Howroyd, Feb 26 2017

Status

approved