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 A215338 Cyclically smooth Lyndon words with 7 colors. 2
 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 (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) * A208776(d). EXAMPLE 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. 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 *) 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, 7)) ); /* necklaces */ /* Lyndon words, via Moebius inversion: */ vl=vector(#vn, n, sumdiv(n, d, moebius(n/d)*vn[d])) CROSSREFS Cf. A208776 (cyclically smooth necklaces, 7 colors). Cf. A215333 (smooth necklaces, 7 colors), A215334 (smooth Lyndon words, 7 colors). Sequence in context: A085964 A281313 A082121 * A176414 A297153 A259168 Adjacent sequences:  A215335 A215336 A215337 * A215339 A215340 A215341 KEYWORD nonn AUTHOR Joerg Arndt, Aug 13 2012 STATUS approved

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Last modified December 14 17:27 EST 2018. Contains 318103 sequences. (Running on oeis4.)