A208775 Number of n-bead necklaces labeled with numbers 1..6 not allowing reversal, with no adjacent beads differing by more than 1.

6, 11, 16, 30, 52, 117, 242, 577, 1360, 3347, 8278, 20978, 53346, 137422, 355978, 928731, 2434580, 6414014, 16961468, 45017417, 119840582, 319916277, 856089572, 2295950281, 6169664562, 16608996492, 44785220118, 120942143132, 327053057574, 885545659155, 2400570958904, 6514679288762, 17697582670400, 48122529680805

Offset

1,1

Links

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

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551 [math.CO], 2008.

Formula

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124699(n). - Andrew Howroyd, Mar 18 2017

Example

All solutions for n=3:
..5....1....1....3....5....5....1....2....2....3....3....6....2....4....4....4
..5....1....2....3....6....5....1....3....2....3....4....6....2....4....5....4
..5....2....2....4....6....6....1....3....2....3....4....6....3....4....5....5

Mathematica

sn[n_, k_] := 1/n*Sum[ Sum[ EulerPhi[j]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/j), {j, Divisors[n]}], {i, 1, k}]; Table[sn[n, 6], {n, 1, 34}] // FullSimplify (* Jean-François Alcover, Oct 31 2017, after Joerg Arndt *)

Prog

(PARI)
/* from the Knopfmacher et al. reference */
default(realprecision, 99); /* using floats */
sn(n, k)=1/n*sum(i=1, k, sumdiv(n, j, eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
vector(66, n, round(sn(n, 6)) )
/* Joerg Arndt, Aug 09 2012 */

Crossrefs

Column 6 of A208777.

Sequence in context: A333533 A191158 A208719 * A242916 A256429 A024730

Adjacent sequences: A208772 A208773 A208774 * A208776 A208777 A208778

Keyword

nonn

Author

R. H. Hardin, Mar 01 2012

Status

approved