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A215335 Cyclically smooth Lyndon words with 3 colors. 2
3, 2, 4, 7, 16, 30, 68, 140, 308, 664, 1476, 3248, 7280, 16286, 36768, 83160, 189120, 431046, 986244, 2261616, 5200776, 11984382, 27676612, 64031520, 148406224, 344500520, 800902564, 1864486560, 4346071600, 10142581552, 23696518916, 55420651440, 129742921992, 304014466080, 712985901856, 1673486122000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

We call a Lyndon word (x[1],x[2],...,x[n]) smooth if abs(x[k]-x[k-1]) <= 1 for 2<=k<=n, and cyclically smooth if abs(x[1]-x[n]) <= 1.

LINKS

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

Latham Boyle, Paul J. Steinhardt, Self-Similar One-Dimensional Quasilattices, arXiv preprint arXiv:1608.08220 [math-ph], 2016.

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

FORMULA

a(n) = sum_{ d divides n } moebius(n/d) * A208772(d).

EXAMPLE

The cyclically smooth necklaces (N) and Lyndon words (L) of length 4 with 3 colors (using symbols ".", "1", and "2") are:

    ....   1       .  N

    ...1   4    ...1  N L

    ..11   4    ..11  N L

    .1.1   2      .1  N

    .111   4    .111  N L

    .121   4    .121  N L

    1111   1       1  N

    1112   4    1112  N L

    1122   4    1122  N L

    1212   2      12  N

    1222   4    1222  N L

    2222   1       2  N

There are 12 necklaces (so A208772(4)=12) and a(4)=7 Lyndon words.

PROG

(PARI)

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)));

vn=vector(66, n, round(sn(n, 3)) ); /* necklaces */

/* Lyndon words, via Moebius inversion: */

vl=vector(#vn, n, sumdiv(n, d, moebius(n/d)*vn[d]))

CROSSREFS

Cf. A208772 (cyclically smooth necklaces, 3 colors).

Cf. A215327 (smooth necklaces, 3 colors), A215328 (smooth Lyndon words, 3 colors).

Sequence in context: A102787 A014193 A128885 * A229976 A084695 A201422

Adjacent sequences:  A215332 A215333 A215334 * A215336 A215337 A215338

KEYWORD

nonn

AUTHOR

Joerg Arndt, Aug 13 2012

STATUS

approved

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Last modified November 18 18:48 EST 2017. Contains 294894 sequences.