A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

1, 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420

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)

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

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

A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 3.

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

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) = 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) = A054661(n) + 3 * A054660(n). - Andrey Zabolotskiy, Dec 17 2020

a(n) = 2 * (A054664(n) + A054660(n)). - Andrey Zabolotskiy, Dec 19 2020

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

Cf. A001037, A027376, A054719, A054660, A054661.

Column k=4 of A074650.

Sequence in context: A079435 A227959 A088015 * A048789 A038069 A274425

Adjacent sequences: A027374 A027375 A027376 * A027378 A027379 A027380

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved