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

0, 1, 3, 7, 13, 25, 44, 78, 136, 242, 422, 747, 1314, 2326, 4121, 7338, 13052, 23288, 41568, 74329, 133011, 238338, 427278, 766652, 1376258, 2472012, 4441916, 7984990, 14358424, 25826779, 46465956, 83616962, 150497816, 270917035, 487753034, 878244512

Offset: 1

Author

Gus Wiseman, May 15 2018

Keywords

• nonn

Comments

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.

Examples

			The a(5) = 13 periodic multisets:
(11), (22), (33), (44),
(111), (222), (333),
(1111), (1122), (1133), (2222), (2233),
(11111).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..500

Crossrefs

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

Programs

Mathematica

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n],GCD@@Length/@Split[#]>1&]],{n,10}]

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

Formula

From Andrew Howroyd, Feb 04 2021: (Start)

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

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

(End)

Extensions

a(12)-a(13) from Robert Price, Sep 15 2018

Terms a(14) and beyond from Andrew Howroyd, Feb 04 2021