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A208773 Number of n-bead necklaces labeled with numbers 1..4 not allowing reversal, with no adjacent beads differing by more than 1. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 15 22:20 EST 2018. Contains 317252 sequences. (Running on oeis4.)