A054718 Number of ternary sequences with primitive period n.

1, 3, 6, 24, 72, 240, 696, 2184, 6480, 19656, 58800, 177144, 530640, 1594320, 4780776, 14348640, 43040160, 129140160, 387400104, 1162261464, 3486725280, 10460350992, 31380882456, 94143178824, 282428998560, 847288609200, 2541864234000, 7625597465304

Offset: 0

Author

N. J. A. Sloane, Apr 20 2000

Keywords

• nonn

Comments

Equivalently, output sequences with primitive period n from a simple cycling shift register.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..2095 (terms 0..650 from Alois P. Heinz)
• E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Crossrefs

Column k=3 of A143324.

Cf. A027375, A054719, A054720, A054721.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(mobius(d)*3^(n/d), d=divisors(n))):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 21 2012

Mathematica

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*3^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

PARI

a(n) = if(n==0,1,sumdiv(n,d, moebius(d) * 3^(n/d) )); \\ Joerg Arndt, Apr 14 2013

Formula

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

a(0) = 1, a(n) = n * A027376(n).

a(n) = 3 * A034741(n).

G.f.: 1 + 3 * Sum_{k>=1} mu(k) * x^k / (1 - 3*x^k). - Ilya Gutkovskiy, Apr 14 2021