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
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
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).
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See 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.
Jake Kettinger, The dynamics of the Hesse derivative on the j-invariant, arXiv:2408.04117 [math.AG], 2024.
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.
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
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
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved