A054719 Number of 4-ary sequences with primitive period n.

1, 4, 12, 60, 240, 1020, 4020, 16380, 65280, 262080, 1047540, 4194300, 16772880, 67108860, 268419060, 1073740740, 4294901760, 17179869180, 68719210560, 274877906940, 1099510578960, 4398046494660, 17592181850100, 70368744177660, 281474959868160

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..1660 (terms 0..500 from Alois P. Heinz)
• E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Crossrefs

Cf. A001868, A027377.

Column k=4 of A143324.

Programs

Maple

A054719 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s); fi; end;

Mathematica

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014 *)

Formula

a(n) = Sum_{d|n} mu(d)*4^(n/d).

a(n) = n * A027377(n), n>0.

G.f.: 1 + 4 * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - Ilya Gutkovskiy, Apr 14 2021