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A208772 Number of n-bead necklaces labeled with numbers 1..3 not allowing reversal, with no adjacent beads differing by more than 1. 4
3, 5, 7, 12, 19, 39, 71, 152, 315, 685, 1479, 3294, 7283, 16359, 36791, 83312, 189123, 431393, 986247, 2262308, 5200851, 11985863, 27676615, 64034954, 148406243, 344507805, 800902879, 1864502926, 4346071603, 10142619039, 23696518919, 55420734752, 129742923475, 304014655205, 712985901943, 1673486556648 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Allowing arbitrary differences between the first and last bead gives A215327. [Joerg Arndt, Aug 08 2012]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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 } A215335(d). - Joerg Arndt, Aug 13 2012

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124696(n). - Andrew Howroyd, Mar 18 2017

EXAMPLE

All solutions for n=4:

..1....2....2....2....1....1....1....3....2....1....2....1

..2....2....3....2....1....2....1....3....3....2....2....1

..1....3....2....2....2....3....1....3....3....2....2....1

..2....3....3....3....2....2....1....3....3....2....2....2

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, 3], {n, 1, 36}] // 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, 3)) )

/* Joerg Arndt, Aug 09 2012 */

CROSSREFS

Column 3 of A208777.

Cf. A215335 (cyclically smooth Lyndon words with 3 colors).

Sequence in context: A241544 A208716 A195821 * A071810 A137576 A161329

Adjacent sequences:  A208769 A208770 A208771 * A208773 A208774 A208775

KEYWORD

nonn

AUTHOR

R. H. Hardin, Mar 01 2012

STATUS

approved

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Last modified November 18 17:56 EST 2017. Contains 294894 sequences.