A208592 Number of n-bead necklaces labeled with numbers -3..3 not allowing reversal, with sum zero.

1, 4, 13, 60, 291, 1564, 8671, 49852, 292927, 1753964, 10656757, 65549844, 407347747, 2553684852, 16130539053, 102563204892, 655918173287, 4216358457772, 27227967629683, 176554882805940, 1149099219084877, 7504110622072860, 49155856119036993, 322903351882566436

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) * A025012(d). - Andrew Howroyd, Mar 02 2017

Example

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

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, 3]; Array[a, 24] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

Crossrefs

Column 3 of A208597.

Sequence in context: A372125 A320359 A057712 * A367888 A351933 A356029

Adjacent sequences: A208589 A208590 A208591 * A208593 A208594 A208595

Keyword

nonn

Author

R. H. Hardin, Feb 29 2012

Extensions

a(20)-a(24) from Andrew Howroyd, Mar 02 2017

Status

approved