A208591 Number of n-bead necklaces labeled with numbers -2..2 not allowing reversal, with sum zero.

1, 3, 7, 23, 77, 297, 1163, 4783, 20041, 85735, 371955, 1634429, 7254547, 32486709, 146576693, 665720447, 3041096613, 13963651875, 64410421307, 298331882891, 1386933722321, 6469566027565, 30271114380919, 142037330038069, 668187211909327

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Formula

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A005191(d). - Andrew Howroyd, Mar 02 2017

Example

All solutions for n=3:
.-2...-1...-2...-2....0...-1...-1
..2...-1....1....0....0....1....0
..0....2....1....2....0....0....1

Mathematica

comps[r_, m_, k_] := Sum[(-1)^i*Binomial[r - 1 - i*m, k - 1]*Binomial[k, i], {i, 0, Floor[(r - k)/m]}]; a[n_Integer, k_] := DivisorSum[n, EulerPhi[n/#] comps[#*(k + 1), 2 k + 1, #] &]/n; a[n_] = a[n, 2]; Array[a, 25] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

Crossrefs

Column 2 of A208597.

Sequence in context: A205481 A341071 A148712 * A104088 A169650 A346771

Adjacent sequences: A208588 A208589 A208590 * A208592 A208593 A208594

Keyword

nonn

Author

R. H. Hardin, Feb 29 2012

Extensions

a(22)-a(25) from Andrew Howroyd, Mar 02 2017

Status

approved