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.
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
KEYWORD
nonn
AUTHOR
STATUS
approved