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

%I A027376 #101 Jan 03 2025 12:32:56
%S A027376 1,3,3,8,18,48,116,312,810,2184,5880,16104,44220,122640,341484,956576,
%T A027376 2690010,7596480,21522228,61171656,174336264,498111952,1426403748,
%U A027376 4093181688,11767874940,33891544368,97764009000,282429535752,817028131140,2366564736720,6863037256208,19924948267224,57906879556410
%N A027376 Number of ternary irreducible monic polynomials of degree n; dimensions of free Lie algebras.
%C A027376 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
%C A027376 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
%C A027376 Number of aperiodic necklaces with n beads of 3 colors. - _Herbert Kociemba_, Nov 25 2016
%C A027376 Number of irreducible harmonic polylogarithms, see page 299 of Gehrmann and Remiddi reference and table 1 of Maître article. - _F. Chapoton_, Aug 09 2021
%C A027376 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
%C A027376 For n>=2, a(n) is the number of orbits of size n of isomorphism classes of elliptic curves under the Hesse derivative, see Theorem 2 of Kettinger link. - _Jake Kettinger_, Aug 07 2024
%D A027376 E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
%D A027376 M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
%H A027376 Seiichi Manyama, <a href="/A027376/b027376.txt">Table of n, a(n) for n = 0..2102</a> (terms 0..200 from T. D. Noe)
%H A027376 Kam Cheong Au, <a href="https://arxiv.org/abs/2007.03957">Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series</a>, arXiv:2007.03957 [math.NT], 2020. See 4th line of Table 1 (p. 6).
%H A027376 Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, <a href="https://arxiv.org/abs/2412.19670">Conjugation, loop and closure invariants of the iterated-integrals signature</a>, arXiv:2412.19670 [math.RA], 2024. See p. 6.
%H A027376 Sayan Dutta, Lorenz Halbeisen, and Norbert Hungerbühler, <a href="https://arxiv.org/abs/2309.05048">Properties of Hesse derivatives of cubic curves</a>, arXiv:2309.05048 [math.AG], 2023.
%H A027376 T. Gehrmann and E. Remiddi, <a href="https://dx.doi.org/10.1016/S0010-4655(02)00139-X">Numerical evaluation of harmonic polylogarithms</a>. Comput. Phys. Comm. 141 (2001), no. 2, 296-312.
%H A027376 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H A027376 Veronika Irvine, <a href="http://hdl.handle.net/1828/7495">Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns</a>, PhD Dissertation, University of Victoria, 2016. See Table A.2.
%H A027376 Jake Kettinger, <a href="https://arxiv.org/abs/2408.04117">The dynamics of the Hesse derivative on the j-invariant</a>, arXiv:2408.04117 [math.AG], 2024.
%H A027376 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A027376 D. Maître, <a href="https://doi.org/10.1016/j.cpc.2005.10.008">HPL, a Mathematica implementation of the harmonic polylogarithms</a>, Computer Physics Communications, Volume 174, Issue 3, 1 February 2006, Pages 222-240.
%H A027376 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H A027376 G. Viennot, <a href="http://dx.doi.org/10.1007/BFb0067950">Algèbres de Lie Libres et Monoïdes Libres</a>, Lecture Notes in Mathematics 691, Springer Verlag, 1978.
%H A027376 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A027376 a(n) = (1/n)*Sum_{d|n} mu(d)*3^(n/d).
%F A027376 (1 - 3*x) = Product_{n>0} (1 - x^n)^a(n).
%F A027376 G.f.: k = 3, 1 - Sum_{i >= 1} mu(i)*log(1 - k*x^i)/i. - _Herbert Kociemba_, Nov 25 2016
%F A027376 a(n) ~ 3^n / n. - _Vaclav Kotesovec_, Jul 01 2018
%F A027376 a(n) = 2*A046211(n) + A046209(n). - _R. J. Mathar_, Oct 21 2021
%e A027376 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
%p A027376 with(numtheory): A027376 := n -> `if`(n = 0, 1,
%p A027376 add(mobius(d)*3^(n/d), d = divisors(n))/n):
%p A027376 seq(A027376(n), n = 0..32);
%t A027376 a[0]=1; a[n_] := Module[{ds=Divisors[n], i}, Sum[MoebiusMu[ds[[i]]]3^(n/ds[[i]]), {i, 1, Length[ds]}]/n]
%t A027376 a[0]=1; a[n_] := DivisorSum[n, MoebiusMu[n/#]*3^#&]/n; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Dec 01 2015 *)
%t A027376 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 *)
%o A027376 (PARI) a(n)=if(n<1,n==0,sumdiv(n,d,moebius(n/d)*3^d)/n)
%Y A027376 Column 3 of A074650.
%Y A027376 Cf. A000031, A001037, A001693, A001867, A027375, A027377, A054718, A102660.
%K A027376 nonn,nice,easy
%O A027376 0,2
%A A027376 _N. J. A. Sloane_