A347862 Total number of polygons left out in all partitions of the set of vertices of a convex n-gon into nonintersecting polygons.

0, 0, 0, 3, 7, 12, 39, 105, 231, 577, 1482, 3549, 8603, 21340, 52122, 126777, 310859, 761199, 1859014, 4549215, 11141085, 27266225, 66760855, 163567911, 400786617, 982265827, 2408361144, 5906499136, 14489105190, 35553445788, 87264949808, 214241203801

Offset: 3

Author

Janaka Rodrigo, Jan 24 2022

Keywords

• nonn

Examples

			a(3) = a(4) = a(5) = 0 since the only partition of the vertices of a triangle, quadrilateral or pentagon into polygons is the full polygon so nothing is left out.
a(6) = 3 since the vertices of a hexagon can be partitioned into two non-intersecting triangles in A350248(6,2) = 3 ways and in each of these cases a quadrilateral is left over.
When partitioning the set of vertices of a convex 13-gon into 1 polygon, the number of polygons remaining is 0.
When partitioning it into 2 polygons, the remaining polygons are 52 quadrilaterals.
When partitioning it into 3 polygons, the remaining polygons are 65 hexagons + 650 quadrilaterals.
When partitioning it into 4 polygons, the remaining polygons are 13 octagons + 117 hexagons + 585 quadrilaterals.
This gives the total as 1482 polygons.
		

Crossrefs

Partitioning into 3 polygons A350116.

Total number of different ways to partition the set of vertices of a convex polygon into nonintersecting polygons A350248.

Programs

PARI

seq(n)={my(p=O(x)); while(serprec(p,x)<=n, p = x + x*y*(1/(1 - x*p^2/(1 - p)) - 1)); Vec(subst(deriv(O(x*x^n) + p^3/(1-p), y), y, 1), 2-n) } \\ Andrew Howroyd, Jan 30 2022

Extensions

More terms from Andrew Howroyd, Jan 30 2022