A054721 Number of 6-ary sequences with primitive period n.

1, 6, 30, 210, 1260, 7770, 46410, 279930, 1678320, 10077480, 60458370, 362797050, 2176734420, 13060694010, 78363884130, 470184976590, 2821108227840, 16926659444730, 101559946544280, 609359740010490, 3656158379595540, 21936950640097710, 131621703479470050

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

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Crossrefs

Column k=6 of A143324.

Programs

Maple

with(numtheory):
a:= n-> `if`(n=0, 1, add(mobius(d)*6^(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]*6^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014 *)

Formula

a(n) = Sum mu(d)*6^(n/d); d|n.

a(0) = 1, a(n) = n * A032164(n).

G.f.: 1 + 6 * Sum_{k>=1} mu(k) * x^k / (1 - 6*x^k). - Ilya Gutkovskiy, Apr 14 2021