A320087 Number of primitive (=aperiodic) ternary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

1, 3, 11, 35, 115, 347, 1075, 3235, 9787, 29387, 88435, 265315, 796755, 2390347, 7173227, 21519947, 64566667, 193700035, 581120523, 1743362283, 5230145947, 15690440099, 47071499707, 141214499227, 423644035627, 1270932113627, 3812797935395, 11438393826035

Offset: 1

Author

Alois P. Heinz, Oct 05 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2096

Crossrefs

Column k=3 of A143327.

Partial sums of A034741.

Cf. A008683, A143325, A143326.

Programs

Maple

b:= n-> add(`if`(d=n, 3^(n-1), -b(d)), d=numtheory[divisors](n)):
a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
seq(a(n), n=1..30);

Mathematica

nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

PARI

a(n) = sum(j=1, n, sumdiv(j, d, 3^(d-1)*moebius(j/d))); \\ Michel Marcus, Dec 11 2020

Formula

a(n) = Sum_{j=1..n} Sum_{d|j} 3^(d-1) * mu(j/d).

a(n) = A143327(n,3).

a(n) = Sum_{j=1..n} A143325(j,3).

a(n) = A143326(n,3) / 3.

G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 3*x^k). - Ilya Gutkovskiy, Dec 11 2020