A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.

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

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

Number of irreducible harmonic polylogarithms, see page 299 of Gehrmann and Remiddi reference and table 1 of Maître article. - F. Chapoton, Aug 09 2021

For n>=2, a(n) is the number of Hesse loops of length 2*n, see Theorem 22 of Dutta, Halbeisen, Hungerbühler link. - Sayan Dutta, Sep 22 2023

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

Seiichi Manyama, Table of n, a(n) for n = 0..2102 (terms 0..200 from T. D. Noe)

Kam Cheong Au, Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series, arXiv:2007.03957 [math.NT], 2020. See 4th line of Table 1 (p. 6).

Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, Properties of Hesse derivatives of cubic curves, arXiv:2309.05048 [math.AG], 2023.

T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms. Comput. Phys. Comm. 141 (2001), no. 2, 296-312.

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.

D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Computer Physics Communications, Volume 174, Issue 3, 1 February 2006, Pages 222-240.

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

a(n) ~ 3^n / n. - Vaclav Kotesovec, Jul 01 2018

a(n) = 2*A046211(n) + A046209(n). - R. J. Mathar, Oct 21 2021

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

with(numtheory): A027376 := n -> `if`(n = 0, 1,
add(mobius(d)*3^(n/d), d = divisors(n))/n):
seq(A027376(n), n = 0..32);

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