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

0,2

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..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.

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

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 *)

Crossrefs

Cf. A001868, A027377.

Column k=4 of A143324.

Sequence in context: A051858 A084709 A057394 * A356662 A356661 A351286

Adjacent sequences: A054716 A054717 A054718 * A054720 A054721 A054722

Keyword

nonn

Author

N. J. A. Sloane, Apr 20 2000

Status

approved