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A208776 Number of n-bead necklaces labeled with numbers 1..7 not allowing reversal, with no adjacent beads differing by more than 1. 4
7, 13, 19, 36, 63, 143, 299, 719, 1711, 4249, 10611, 27144, 69727, 181467, 475147, 1253475, 3324103, 8862889, 23729747, 63791064, 172066959, 465577215, 1263208683, 3435919395, 9366558151, 25585896137, 70019831931, 191943278804, 526978629663, 1448862872667, 3988658225035, 10993823704779, 30335737469495, 83793424341677 (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 } A215338(d). - Joerg Arndt, Aug 13 2012

a(n) = (1/n) * Sum_{d | n} totient(n/d) * A124700(n). - Andrew Howroyd, Mar 18 2017

EXAMPLE

All solutions for n=3:

..3....2....6....4....4....1....5....2....2....6....3....3....5....6....5....1

..3....3....6....4....4....1....5....2....2....7....4....3....6....6....5....2

..4....3....6....5....4....1....5....3....2....7....4....3....6....7....6....2

..

..4....7....1

..5....7....1

..5....7....2

MATHEMATICA

sn[n_, k_] := 1/n*Sum[ DivisorSum[n, EulerPhi[#]*(1 + 2*Cos[i*Pi/(k + 1)])^(n/#) &], {i, 1, k}]; Table[sn[n, 7], {n, 1, 34}] // 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, 7)) )

/* Joerg Arndt, Aug 09 2012 */

CROSSREFS

Column 7 of A208777.

Cf. A215338 (cyclically smooth Lyndon words with 7 colors).

Sequence in context: A048375 A198035 A208720 * A108295 A071923 A048646

Adjacent sequences:  A208773 A208774 A208775 * A208777 A208778 A208779

KEYWORD

nonn

AUTHOR

R. H. Hardin, Mar 01 2012

STATUS

approved

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Last modified November 23 19:03 EST 2017. Contains 295128 sequences.