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 A032164 Number of aperiodic necklaces of n beads of 6 colors; dimensions of free Lie algebras. 8
 1, 6, 15, 70, 315, 1554, 7735, 39990, 209790, 1119720, 6045837, 32981550, 181394535, 1004668770, 5597420295, 31345665106, 176319264240, 995685849690, 5642219252460, 32071565263710, 182807918979777 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Petros Hadjicostas, Aug 31 2018: (Start) For each m >= 1, the CHK[m] transform of sequence (c(n): n>=1) has generating function B_m(x) = (1/m)*Sum_{d|m} mu(d)*C(x^d)^{m/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n >= 1). As a result, the CHK transform of sequence (c(n): n >= 1) has generating function B(x) = Sum_{m >= 1} B_m(x) = -Sum_{n >= 1} (mu(n)/n)*log(1 - C(x^n)). For n, k >= 1, let a_k(n) = number of aperiodic necklaces of n beads of k colors. We then have (a_k(n): n >= 1) = CHK(c_k(n): n >= 1), where c_k(1) = k and c_k(n) = 0 for all n >= 2, with g.f. C_k(x) = Sum_{n >= 1} c_k(n)*x^n = k*x. The g.f. of (a_k(n): n >= 1) is A_k(x) = Sum_{n >= 1} a_k(n)*x^n = -Sum_{n >= 1} (mu(n)/n)*log(1-k*x^n), which is Herbert Kociemba's general formula below (except for the initial term a_k(0) = 1). For the current sequence, k = 6. (End) REFERENCES M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1289 (terms 0..200 from T. D. Noe) C. G. Bower, Transforms (2) E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only] G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978. FORMULA "CHK" (necklace, identity, unlabeled) transform of 6, 0, 0, 0... a(n) = Sum_{d|n} mu(d)*6^(n/d)/n, for n>0. G.f.: k=6, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016 MATHEMATICA f[d_] := MoebiusMu[d]*6^(n/d)/n; a[n_] := Total[f /@ Divisors[n]]; a = 1; Table[a[n], {n, 0, 20}](* Jean-François Alcover, Nov 07 2011 *) mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 6], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *) PROG (PARI) a(n) = if (n==0, 1, sumdiv(n, d, moebius(d)*6^(n/d)/n)); \\ Michel Marcus, Dec 01 2015 CROSSREFS Column 6 of A074650. Cf. A001037, A001692 (5 colors). Cf. A054721. Sequence in context: A324973 A318555 A035077 * A177122 A108540 A232170 Adjacent sequences:  A032161 A032162 A032163 * A032165 A032166 A032167 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified June 22 21:29 EDT 2021. Contains 345393 sequences. (Running on oeis4.)