A356366 Number of (directed) circuits in the complete undirected graph on n labeled vertices.

1, 2, 5, 18, 523, 44884, 227838935, 1086696880188, 1566338449874827101, 694432397394116143569646

Offset: 1

Author

Max Alekseyev, Oct 16 2022

Keywords

• nonn
• more
• walk

Comments

In other words, number of closed trails up to cyclic rotations (cf. A357855).

Examples

			For n = 3, we have 5 circuits: 3 of length 0 (singleton vertices), and 2 of length 3 (1->2->3->1 and 1->3->2->1).
		

Crossrefs

Cf. A007082, A135388, A232545, A350028, A357855, A357856, A357857, A357885, A357886, A357887.

Formula

a(n) = Sum_{k = 0..n(n-1)/2} A357887(n,k) = n + n * Sum_{k = 1..n(n-1)/2} A357885(n,k) / k.

Extensions

a(9) from Bert Dobbelaere, Oct 17 2022

a(10) from Max Alekseyev, Jul 17 2025