A208773 Number of n-bead necklaces labeled with numbers 1..4 not allowing reversal, with no adjacent beads differing by more than 1.

4, 7, 10, 18, 30, 65, 128, 293, 658, 1544, 3622, 8711, 20924, 50889, 124150, 304718, 750334, 1855429, 4600696, 11442853, 28528618, 71294416, 178529670, 447923761, 1125756860, 2833917147, 7144466842, 18036449390, 45591671454, 115381885423, 292329164912, 741411257693, 1882219950046, 4782783122992, 12163730636250

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:0809.0551v1 [math.CO], 2008.

Formula

a(n) = Sum_{ d | n } A215336(d). - Joerg Arndt, Aug 13 2012

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124697(n). - Andrew Howroyd, Mar 18 2017

Example

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

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, 4], {n, 1, 35}] // 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, 4)) )
/* Joerg Arndt, Aug 09 2012 */

Crossrefs

Column 4 of A208777.

Cf. A215336 (cyclically smooth Lyndon words with 4 colors).

Sequence in context: A310719 A103408 A208717 * A088408 A027752 A161334

Adjacent sequences: A208770 A208771 A208772 * A208774 A208775 A208776

Keyword

nonn

Author

R. H. Hardin, Mar 01 2012

Status

approved