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A304648 Number of different periodic multisets that fit within some normal multiset of weight n.

%I A304648 #14 Feb 04 2021 16:31:09
%S A304648 0,1,3,7,13,25,44,78,136,242,422,747,1314,2326,4121,7338,13052,23288,
%T A304648 41568,74329,133011,238338,427278,766652,1376258,2472012,4441916,
%U A304648 7984990,14358424,25826779,46465956,83616962,150497816,270917035,487753034,878244512
%N A304648 Number of different periodic multisets that fit within some normal multiset of weight n.
%C A304648 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 A304648 Andrew Howroyd, <a href="/A304648/b304648.txt">Table of n, a(n) for n = 1..500</a>
%F A304648 From _Andrew Howroyd_, Feb 04 2021: (Start)
%F A304648 a(n) = A027941(n) - A303976(n).
%F A304648 G.f.: Sum_{d>=2} -mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)).
%F A304648 (End)
%e A304648 The a(5) = 13 periodic multisets:
%e A304648 (11), (22), (33), (44),
%e A304648 (111), (222), (333),
%e A304648 (1111), (1122), (1133), (2222), (2233),
%e A304648 (11111).
%t A304648 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A304648 Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]
%o A304648 (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 A304648 Cf. A000217, A001597, A018783, A027941, A178472, A210554, A303547, A303709, A303974, A303976.
%K A304648 nonn
%O A304648 1,3
%A A304648 _Gus Wiseman_, May 15 2018
%E A304648 a(12)-a(13) from _Robert Price_, Sep 15 2018
%E A304648 Terms a(14) and beyond from _Andrew Howroyd_, Feb 04 2021