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%I #28 Apr 14 2021 22:25:16
%S 1,4,12,60,240,1020,4020,16380,65280,262080,1047540,4194300,16772880,
%T 67108860,268419060,1073740740,4294901760,17179869180,68719210560,
%U 274877906940,1099510578960,4398046494660,17592181850100,70368744177660,281474959868160
%N Number of 4-ary sequences with primitive period n.
%C Equivalently, output sequences with primitive period n from a simple cycling shift register.
%H Seiichi Manyama, <a href="/A054719/b054719.txt">Table of n, a(n) for n = 0..1660</a> (terms 0..500 from Alois P. Heinz)
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F a(n) = Sum_{d|n} mu(d)*4^(n/d).
%F a(n) = n * A027377(n), n>0.
%F G.f.: 1 + 4 * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - _Ilya Gutkovskiy_, Apr 14 2021
%p 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;
%t 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 *)
%Y Cf. A001868, A027377.
%Y Column k=4 of A143324.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Apr 20 2000
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