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

3,4

Links

Table of n, a(n) for n=3..34.

Example

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.

Prog

(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

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.

Sequence in context: A167490 A081533 A096856 * A345260 A108154 A346154

Adjacent sequences: A347859 A347860 A347861 * A347863 A347864 A347865

Keyword

nonn

Author

Janaka Rodrigo, Jan 24 2022

Extensions

More terms from Andrew Howroyd, Jan 30 2022

Status

approved