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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

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.

G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.

Index entries for sequences related to Lyndon words

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[0] = 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: A233450 A298374 A035077 * A177122 A108540 A232170

Adjacent sequences:  A032161 A032162 A032163 * A032165 A032166 A032167

KEYWORD

nonn,easy,nice

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 20 10:26 EST 2018. Contains 299385 sequences. (Running on oeis4.)