A304648 - OEIS

%I #14 Feb 04 2021 16:31:09

%S 0,1,3,7,13,25,44,78,136,242,422,747,1314,2326,4121,7338,13052,23288,

%T 41568,74329,133011,238338,427278,766652,1376258,2472012,4441916,

%U 7984990,14358424,25826779,46465956,83616962,150497816,270917035,487753034,878244512

%N Number of different periodic multisets that fit within some normal multiset of weight n.

%C A multiset is normal if it spans an initial interval of positive integers. It is periodic if its multiplicities have a common divisor greater than 1.

%H Andrew Howroyd, <a href="/A304648/b304648.txt">Table of n, a(n) for n = 1..500</a>

%F From _Andrew Howroyd_, Feb 04 2021: (Start)

%F a(n) = A027941(n) - A303976(n).

%F G.f.: Sum_{d>=2} -mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)).

%F (End)

%e The a(5) = 13 periodic multisets:

%e (11), (22), (33), (44),

%e (111), (222), (333),

%e (1111), (1122), (1133), (2222), (2233),

%e (11111).

%t allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];

%t Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]

%o (PARI) seq(n)=Vec(sum(d=2, n, -moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x), -n) \\ _Andrew Howroyd_, Feb 04 2021

%Y Cf. A000217, A001597, A018783, A027941, A178472, A210554, A303547, A303709, A303974, A303976.

%K nonn

%O 1,3

%A _Gus Wiseman_, May 15 2018

%E a(12)-a(13) from _Robert Price_, Sep 15 2018

%E Terms a(14) and beyond from _Andrew Howroyd_, Feb 04 2021