A361285 Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are allowed.

0, 0, 1, 10, 85, 695, 5600, 45080, 364854, 2973270, 24382875, 200967250, 1662197251, 13772638789, 114126098450, 944285871200, 7791140945180, 64038240953196, 523977421054245, 4266101869823850, 34554155058753505, 278417272387723315, 2231755184899383220, 17799741659621513240

Offset

1,4

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Links

Table of n, a(n) for n=1..24.

Ivaylo Kortezov, Sets of Paths between Vertices of a Polygon, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.

Formula

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) + 9*5^(n-3) + 3^n + 27).

E.g.f.: x^3*exp(x)*(exp(2*x) + 3)^3/384. - Andrew Howroyd, Mar 07 2023

Example

a(4) = A360021(4) + 4*A360021(3) = 6 + 4 = 10 since either all the 4 points are used or one is not.

Prog

(PARI) a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) + 9*5^(n-3) + 3^n + 27)} \\ Andrew Howroyd, Mar 07 2023

Crossrefs

If there is only one path, we get A360715. If there is are two paths, we get A360717. If all n points need to be used, we get A360021.

Sequence in context: A291389 A081903 A346319 * A144639 A233667 A038235

Adjacent sequences: A361282 A361283 A361284 * A361286 A361287 A361288

Keyword

nonn,easy

Author

Ivaylo Kortezov, Mar 07 2023

Status

approved