A362786 Number of unordered triples of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon.

0, 0, 0, 5, 63, 476, 2772, 13680, 60060, 241472, 906048, 3214848, 10890880, 35481600, 111794176, 342171648, 1021031424, 2979102720, 8520171520, 23934468096, 66156625920, 180198047744, 484304486400, 1285790105600, 3375480176640, 8769899593728, 22567515586560, 57557594931200

Offset: 3

Author

Ivaylo Kortezov, May 04 2023

Keywords

• nonn

Examples

			For n=7 we have one 3-node path and two 2-node paths. Call two paths adjacent if we can choose one node from each path so that the two nodes are adjacent vertices of the n-gon. Then either each pair of paths is adjacent, or the two 2-node paths are not adjacent, or a 2-node path is not adjacent to the 3-node path. In each of these three cases there are 7 choices for the set of nodes for the 3-node path and 3 ways to connect them, and then the 2-node paths are uniquely determined. Thus a(7) = 3*7*3 = 63.
		

Crossrefs

The number of unordered pairs of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon is A308914(n). The number of unordered triples of (not necessarily disjoint) self-avoiding paths with nodes that cover all vertices of a convex n-gon is A359404(n).

Formula

a(n) = 2^(n-12)*n*(n-1)*(n-2)*(n-4)*(n-5)*(n+2)*(n+9)/90 for n > 3; 0 for n=3.