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A263659 Number of (0, 1)-necklaces of length n without zigzags (see reference for precise definition). 5
0, 2, 2, 2, 3, 4, 5, 6, 8, 10, 15, 20, 31, 42, 64, 94, 143, 212, 329, 494, 766, 1170, 1811, 2788, 4341, 6714, 10462, 16274, 25415, 39652, 62075, 97110, 152288, 238838, 375167, 589528, 927555, 1459962, 2300348, 3626242, 5721045 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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..200

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) * A007039(d). - Andrew Howroyd, Feb 26 2017

EXAMPLE

For n=5 the necklaces are 00000, 11111, 00011, 00111 so a(5)=4.

MATHEMATICA

(* b = A007039 *) b[n_ /; n<4] = 2; b[4] = 6; b[n_] := b[n] = 2*b[n-1] - b[n-2] + b[n-4];

a[0] = 0; a[n_] := (1/n) * DivisorSum[n, EulerPhi[n/#] * b[#]&];

Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

CROSSREFS

Antidiagonal sums of A263657.

Cf. A007039, A263655, A263656, A263658.

Sequence in context: A026837 A005855 A096748 * A022866 A099388 A193941

Adjacent sequences:  A263656 A263657 A263658 * A263660 A263661 A263662

KEYWORD

nonn

AUTHOR

Felix Fröhlich, Oct 23 2015

EXTENSIONS

a(25)-a(40) from Andrew Howroyd, Feb 26 2017

STATUS

approved

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Last modified May 17 14:30 EDT 2021. Contains 343972 sequences. (Running on oeis4.)