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

Author

Felix Fröhlich, Oct 23 2015

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Main diagonal of A263657.

Cf. A007039, A263655, A263656, A263659.

Programs

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 *)

Formula

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

Extensions

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