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A215337 Cyclically smooth Lyndon words with 5 colors. 2
5, 4, 8, 15, 36, 74, 180, 411, 996, 2400, 5940, 14707, 36972, 93276, 237264, 606030, 1556028, 4009118, 10367892, 26888925, 69930264, 182296212, 476262756, 1246695079, 3269321352, 8587452204, 22590645408, 59510993607, 156973954860, 414552239458, 1096017973380, 2900753084400, 7684758670248, 20377460964156, 54081265456116 (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

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) * A208774(d).

EXAMPLE

The cyclically smooth necklaces (N) and Lyndon words (L) of length 4 with 5 colors (using symbols ".", "1", "2", "3", and "4") 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

    1232   4    1232  N L

    2222   1       2  N

    2223   4    2223  N L

    2233   4    2233  N L

    2323   2      23  N

    2333   4    2333  N L

    2343   4    2343  N L

    3333   1       3  N

    3334   4    3334  N L

    3344   4    3344  N L

    3434   2      34  N

    3444   4    3444  N L

    4444   1       4  N

There are 24 necklaces (so A208774(4)=24) and a(4)=15 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, 5)) ); /* necklaces */

/* Lyndon words, via Moebius inversion: */

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

CROSSREFS

Sequence in context: A187891 A086407 A160427 * A090124 A097943 A241420

Adjacent sequences:  A215334 A215335 A215336 * A215338 A215339 A215340

KEYWORD

nonn

AUTHOR

Joerg Arndt, Aug 13 2012

STATUS

approved

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