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

1,3

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.

Links

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

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)

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A259343 A210613 A363707 * A243737 A269832 A341579

Adjacent sequences: A304645 A304646 A304647 * A304649 A304650 A304651

Keyword

nonn

Author

Gus Wiseman, May 15 2018

Extensions

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

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

Status

approved