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

0,2

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.

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

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 *)

Prog

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

Crossrefs

Column k=3 of A143324.

Cf. A027375, A054719, A054720, A054721.

Sequence in context: A148655 A148656 A279300 * A132390 A363016 A327643

Adjacent sequences: A054715 A054716 A054717 * A054719 A054720 A054721

Keyword

nonn

Author

N. J. A. Sloane, Apr 20 2000

Status

approved