cp's OEIS Frontend

A320088 Number of primitive (=aperiodic) 4-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

%I A320088 #11 Dec 11 2020 06:15:04
%S A320088 1,4,19,79,334,1339,5434,21754,87274,349159,1397734,5590954,22368169,
%T A320088 89472934,357908119,1431633559,5726600854,22906403494,91625880229,
%U A320088 366503524969,1466015148634,5864060611159,23456246655574,93824986622614,375299963333014,1501199853398419
%N A320088 Number of primitive (=aperiodic) 4-ary words with length less than or equal to n which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.
%H A320088 Alois P. Heinz, <a href="/A320088/b320088.txt">Table of n, a(n) for n = 1..1661</a>
%F A320088 a(n) = Sum_{j=1..n} Sum_{d|j} 4^(d-1) * mu(j/d).
%F A320088 a(n) = A143327(n,4).
%F A320088 a(n) = Sum_{j=1..n} A143325(j,4).
%F A320088 a(n) = A143326(n,4) / 4.
%F A320088 G.f.: (1/(1 - x)) * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - _Ilya Gutkovskiy_, Dec 11 2020
%p A320088 b:= n-> add(`if`(d=n, 4^(n-1), -b(d)), d=numtheory[divisors](n)):
%p A320088 a:= proc(n) option remember; b(n)+`if`(n<2, 0, a(n-1)) end:
%p A320088 seq(a(n), n=1..30);
%t A320088 nmax = 20; Rest[CoefficientList[Series[1/(1-x) * Sum[MoebiusMu[k] * x^k / (1 - 4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Dec 11 2020 *)
%o A320088 (PARI) a(n) = sum(j=1, n, sumdiv(j, d, 4^(d-1)*moebius(j/d))); \\ _Michel Marcus_, Dec 11 2020
%Y A320088 Column k=4 of A143327.
%Y A320088 Partial sums of A295505.
%Y A320088 Cf. A008683, A143325, A143326.
%K A320088 nonn
%O A320088 1,2
%A A320088 _Alois P. Heinz_, Oct 05 2018