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A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras. 30
1, 3, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720, 6863037256208, 19924948267224, 57906879556410 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of Lyndon words of length n on {1,2,3}. A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts. - John W. Layman, Jan 24 2006

Exponents in an expansion of the Hardy-Littlewood constant product(1-(3*p-1)/(p-1)^3, p prime >= 5), whose decimal expansion is in A065418: the constant equals product_{n>=2} (zeta(n)*(1-2^-n)*(1-3^-n))^-a(n). - Michael Somos, Apr 05 2003

Number of aperiodic necklaces with n beads of 3 colors. - Herbert Kociemba, Nov 25 2016

REFERENCES

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

LINKS

T. D. Noe, Table of n, a(n) for n=0..200

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016. See Table A.2.

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

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

a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d).

(1-3*x) = Product_{n>0} (1-x^n)^a(n).

G.f.: k=3, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

EXAMPLE

For n = 2 the a(2)=3 polynomials are  x^2+1, x^2+x+2, x^2+2*x+2. - Robert Israel, Dec 16 2015

MAPLE

A027376 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*3^(n/d); od; RETURN(s/n); fi; end;

MATHEMATICA

a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n]

a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 01 2015 *)

mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 3], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *)

PROG

(PARI) a(n)=if(n<1, n==0, sumdiv(n, d, moebius(n/d)*3^d)/n)

CROSSREFS

Column 3 of A074650.

Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660.

Sequence in context: A059197 A049974 A049972 * A190659 A202536 A038068

Adjacent sequences:  A027373 A027374 A027375 * A027377 A027378 A027379

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 19 08:51 EST 2017. Contains 294923 sequences.