A215338 Cyclically smooth Lyndon words with 7 colors.
7, 6, 12, 23, 56, 118, 292, 683, 1692, 4180, 10604, 26978, 69720, 181162, 475072, 1252756, 3324096, 8861054, 23729740, 63786792, 172066648, 465566598, 1263208676, 3435891568, 9366558088, 25585826404, 70019830220, 191943097314, 526978629656, 1448862393216, 3988658225028, 10993822451304, 30335737458872, 83793421017568
Offset: 1
Keywords
Examples
The cyclically smooth necklaces (N) and Lyndon words (L) of length 4 with 7 colors (using symbols ".", "1", "2", "3", "4", "5", and "6") 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
3454 4 3454 N L
4444 1 4 N
4445 4 4445 N L
4455 4 4455 N L
4545 2 45 N
4555 4 4555 N L
4565 4 4565 N L
5555 1 5 N
5556 4 5556 N L
5566 4 5566 N L
5656 2 56 N
5666 4 5666 N L
6666 1 6 N
There are 36 necklaces (so A208776(4)=36) and a(4)=23 Lyndon words.
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.
Crossrefs
Programs
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Mathematica
terms = 40; sn[n_, k_] := 1/n Sum[EulerPhi[j] (1+2Cos[i Pi/(k+1)])^(n/j), {i, 1, k}, {j, Divisors[n]}]; vn = Table[Round[sn[n, 7]], {n, terms}]; vl = Table[Sum[MoebiusMu[n/d] vn[[d]], {d, Divisors[n]}], {n, terms}] (* Jean-François Alcover, Jul 22 2018, after Joerg Arndt *) -
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,7)) ); /* necklaces */ /* Lyndon words, via Moebius inversion: */ vl=vector(#vn,n, sumdiv(n,d, moebius(n/d)*vn[d]))
Formula
a(n) = sum_{ d divides n } moebius(n/d) * A208776(d).
Comments