cp's OEIS Frontend

A215338 Cyclically smooth Lyndon words with 7 colors.

%I A215338 #12 Jul 22 2018 10:43:11
%S A215338 7,6,12,23,56,118,292,683,1692,4180,10604,26978,69720,181162,475072,
%T A215338 1252756,3324096,8861054,23729740,63786792,172066648,465566598,
%U A215338 1263208676,3435891568,9366558088,25585826404,70019830220,191943097314,526978629656,1448862393216,3988658225028,10993822451304,30335737458872,83793421017568
%N A215338 Cyclically smooth Lyndon words with 7 colors.
%C A215338 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 A215338 Vincenzo Librandi, <a href="/A215338/b215338.txt">Table of n, a(n) for n = 1..200</a>
%H A215338 Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.
%F A215338 a(n) = sum_{ d divides n } moebius(n/d) * A208776(d).
%e A215338 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:
%e A215338     ....   1       .  N
%e A215338     ...1   4    ...1  N L
%e A215338     ..11   4    ..11  N L
%e A215338     .1.1   2      .1  N
%e A215338     .111   4    .111  N L
%e A215338     .121   4    .121  N L
%e A215338     1111   1       1  N
%e A215338     1112   4    1112  N L
%e A215338     1122   4    1122  N L
%e A215338     1212   2      12  N
%e A215338     1222   4    1222  N L
%e A215338     1232   4    1232  N L
%e A215338     2222   1       2  N
%e A215338     2223   4    2223  N L
%e A215338     2233   4    2233  N L
%e A215338     2323   2      23  N
%e A215338     2333   4    2333  N L
%e A215338     2343   4    2343  N L
%e A215338     3333   1       3  N
%e A215338     3334   4    3334  N L
%e A215338     3344   4    3344  N L
%e A215338     3434   2      34  N
%e A215338     3444   4    3444  N L
%e A215338     3454   4    3454  N L
%e A215338     4444   1       4  N
%e A215338     4445   4    4445  N L
%e A215338     4455   4    4455  N L
%e A215338     4545   2      45  N
%e A215338     4555   4    4555  N L
%e A215338     4565   4    4565  N L
%e A215338     5555   1       5  N
%e A215338     5556   4    5556  N L
%e A215338     5566   4    5566  N L
%e A215338     5656   2      56  N
%e A215338     5666   4    5666  N L
%e A215338     6666   1       6  N
%e A215338 There are 36 necklaces (so A208776(4)=36) and a(4)=23 Lyndon words.
%t A215338 terms = 40;
%t A215338 sn[n_, k_] := 1/n Sum[EulerPhi[j] (1+2Cos[i Pi/(k+1)])^(n/j), {i, 1, k}, {j, Divisors[n]}];
%t A215338 vn = Table[Round[sn[n, 7]], {n, terms}];
%t A215338 vl = Table[Sum[MoebiusMu[n/d] vn[[d]], {d, Divisors[n]}], {n, terms}] (* _Jean-François Alcover_, Jul 22 2018, after _Joerg Arndt_ *)
%o A215338 (PARI)
%o A215338 default(realprecision,99); /* using floats */
%o A215338 sn(n,k)=1/n*sum(i=1,k,sumdiv(n,j,eulerphi(j)*(1+2*cos(i*Pi/(k+1)))^(n/j)));
%o A215338 vn=vector(66,n, round(sn(n,7)) ); /* necklaces */
%o A215338 /* Lyndon words, via Moebius inversion: */
%o A215338 vl=vector(#vn,n, sumdiv(n,d, moebius(n/d)*vn[d]))
%Y A215338 Cf. A208776 (cyclically smooth necklaces, 7 colors).
%Y A215338 Cf. A215333 (smooth necklaces, 7 colors), A215334 (smooth Lyndon words, 7 colors).
%K A215338 nonn
%O A215338 1,1
%A A215338 _Joerg Arndt_, Aug 13 2012