A027380 Number of irreducible polynomials of degree n over GF(8); dimensions of free Lie algebras.

1, 8, 28, 168, 1008, 6552, 43596, 299592, 2096640, 14913024, 107370900, 780903144, 5726600880, 42288908760, 314146029564, 2345624803704, 17592184995840, 132458812569720, 1000799909722368, 7585009898729256

Offset

0,2

Comments

Number of Lyndon words with 8 letters. - Joerg Arndt, Jul 29 2014

Number of aperiodic necklaces with n beads of 8 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..1110 (terms 0..200 from T. D. Noe)

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

G.f.: k=8, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

a(n) = Sum_{d|n} mu(d)*8^(n/d)/n for n > 0. - Andrew Howroyd, Oct 13 2017

Example

G.f. = 1 + 8*x + 28*x^2 + 168*x^3 + 1008*x^4 + 6552*x^5 + 43596*x^6 + ...

Maple

A027380 := proc(n)
    local d;
    if n = 0 then
        1;
    else
        add( 8^(n/d)*numtheory[mobius](d), d=numtheory[divisors](n)) ;
        %/n ;
    end if;
end proc: # R. J. Mathar, Jun 09 2016

Mathematica

f[n_] := (1/n)*Sum[MoebiusMu[d]*8^(n/d), {d, Divisors[n]}]; f[0] = 1; Array[f, 20, 0] (* Robert G. Wilson v, Jul 28 2014 *)
mx=40; f[x_, k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i, {i, 1, mx}]; CoefficientList[Series[f[x, 8], {x, 0, mx}], x] (* Herbert Kociemba, Nov 25 2016 *)

Prog

(PARI) a(n) = if(n, sumdiv(n, d, moebius(d)*8^(n/d))/n, 1) \\ Altug Alkan, Dec 01 2015

Crossrefs

Column 8 of A074650.

Cf. A001037, A001693.

Sequence in context: A200188 A255276 A110046 * A226744 A058289 A152014

Adjacent sequences: A027377 A027378 A027379 * A027381 A027382 A027383

Keyword

nonn

Author

N. J. A. Sloane

Status

approved