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

1,2

Links

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

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

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 *)

Prog

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

Crossrefs

Column k=3 of A143327.

Partial sums of A034741.

Cf. A008683, A143325, A143326.

Sequence in context: A125672 A107683 A259400 * A014335 A147474 A247417

Adjacent sequences: A320084 A320085 A320086 * A320088 A320089 A320090

Keyword

nonn

Author

Alois P. Heinz, Oct 05 2018

Status

approved