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A215336 Cyclically smooth Lyndon words with 4 colors.

%I A215336 #13 Jul 22 2018 10:44:25
%S A215336 4,3,6,11,26,52,124,275,648,1511,3618,8635,20920,50758,124114,304425,
%T A215336 750330,1854716,4600692,11441298,28528484,71290791,178529666,
%U A215336 447914775,1125756830,2833896220,7144466184,18036398490,45591671450,115381759707,292329164908,741410952975,1882219946418,4782782372655,12163730636096
%N A215336 Cyclically smooth Lyndon words with 4 colors.
%C A215336 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.
%H A215336 Vincenzo Librandi, <a href="/A215336/b215336.txt">Table of n, a(n) for n = 1..200</a>
%H A215336 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551 [math.CO], 2008.
%F A215336 a(n) = sum_{ d divides n } moebius(n/d) * A208773(d).
%e A215336 The cyclically smooth necklaces (N) and Lyndon words (L) of length 4 with 4 colors (using symbols ".", "1", "2", and "3") are:
%e A215336     ....   1       .  N
%e A215336     ...1   4    ...1  N L
%e A215336     ..11   4    ..11  N L
%e A215336     .1.1   2      .1  N
%e A215336     .111   4    .111  N L
%e A215336     .121   4    .121  N L
%e A215336     1111   1       1  N
%e A215336     1112   4    1112  N L
%e A215336     1122   4    1122  N L
%e A215336     1212   2      12  N
%e A215336     1222   4    1222  N L
%e A215336     1232   4    1232  N L
%e A215336     2222   1       2  N
%e A215336     2223   4    2223  N L
%e A215336     2233   4    2233  N L
%e A215336     2323   2      23  N
%e A215336     2333   4    2333  N L
%e A215336     3333   1       3  N
%e A215336 There are 18 necklaces (so A208773(4)=24) and a(4)=11 Lyndon words.
%t A215336 terms = 40;
%t A215336 sn[n_, k_] := 1/n Sum[EulerPhi[j] (1+2Cos[i Pi/(k+1)])^(n/j), {i, 1, k}, {j, Divisors[n]}];
%t A215336 vn = Table[Round[sn[n, 4]], {n, terms}];
%t A215336 vl = Table[Sum[MoebiusMu[n/d] vn[[d]], {d, Divisors[n]}], {n, terms}] (* _Jean-François Alcover_, Jul 22 2018, after _Joerg Arndt_ *)
%o A215336 (PARI)
%o A215336 default(realprecision,99); /* using floats */
%o A215336 sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
%o A215336 vn=vector(66,n, round(sn(n,4)) ); /* necklaces */
%o A215336 /* Lyndon words, via Moebius inversion: */
%o A215336 vl=vector(#vn,n, sumdiv(n,d, moebius(n/d)*vn[d]))
%Y A215336 Cf. A208773 (cyclically smooth necklaces, 4 colors).
%Y A215336 Cf. A215329 (smooth necklaces, 4 colors), A215330 (smooth Lyndon words, 4 colors).
%K A215336 nonn
%O A215336 1,1
%A A215336 _Joerg Arndt_, Aug 13 2012