OFFSET
0,2
COMMENTS
Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).
Number of aperiodic necklaces with n beads of 4 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
Seiichi Manyama, Table of n, a(n) for n = 0..1666 (terms 0..200 from T. D. Noe)
Daniel Gabric and Joe Sawada, Necklaces and Lyndon words in colexicographic order, arXiv:2607.05324 [math.CO], 2026.
E. N. Gilbert and John Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
G. Niklasch, Some number theoretical constants: 1000-digit values, 2002. [Cached copy]
Apisit Pakapongpun and Thomas Ward, Functorial Orbit counting, J. Int. Seq. 12 (2009), Art. 09.2.4. See Example 3.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Int. Seq. 4 (2001), Art. 01.2.1.
Gérard Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lect. Notes Math. 691, Springer Verlag 1978.
FORMULA
a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) = A054719(n)/n, n>0. - R. J. Mathar, Dec 16 2024
MAPLE
A027377 := proc(n) local d, s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;
MATHEMATICA
a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* Jean-François Alcover, Nov 29 2011 *)
mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 4], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *)
PROG
(PARI) a(n)=if(n, sumdiv(n, d, moebius(d)<<(2*n/d))/n, 1) \\ Charles R Greathouse IV, Nov 29 2011
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved