A208538 Number of n-bead necklaces of n colors allowing reversal, with no adjacent beads having the same color.

1, 1, 1, 21, 102, 1505, 19995, 365260, 7456596, 174489813, 4545454545, 130773238871, 4115123283810, 140620807064413, 5185603185296625, 205262771447683860, 8680820740569200760, 390641235316599920745, 18637772246193096746253, 939749336469457562916217

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..80

Formula

a(2n+1) = A208533(2n+1)/2 for n > 0, a(2n) = (A208533(2n) + n*(2n-1)^n)/2. - Andrew Howroyd, Mar 12 2017

Example

All solutions for n=4:
..1....1....1....1....2....1....1....1....2....1....1....3....2....2....1....1
..2....4....3....2....3....2....3....3....4....3....2....4....3....4....2....2
..4....2....2....3....2....4....1....4....2....2....1....3....2....3....1....1
..2....4....4....2....3....3....3....3....4....3....3....4....4....4....2....4
..
..1....1....2....1....1
..3....4....3....2....4
..1....3....4....3....1
..4....4....3....4....4

Mathematica

T[n_, k_] := If[n == 1, k, (DivisorSum[n, EulerPhi[n/#]*(k - 1)^# &]/n + If[OddQ[n], 1 - k, k*(k - 1)^(n/2)/2])/2]; a[n_] = T[n, n]; Array[a, 20] (* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)

Crossrefs

Diagonal of A208544.

Sequence in context: A044272 A044653 A076454 * A303681 A304132 A304220

Adjacent sequences: A208535 A208536 A208537 * A208539 A208540 A208541

Keyword

nonn

Author

R. H. Hardin, Feb 27 2012

Extensions

a(12)-a(20) from Andrew Howroyd, Mar 12 2017

Status

approved