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