OFFSET

1,1

COMMENTS

For vertices not on the boundary, the number of polygons meeting at a vertex is simply the degree (or valency) of that vertex.

Row sums are A331755.

Sum_k k*T(n,k) gives A333276.

See A333275 for the degrees of the non-boundary vertices.

Row n is the sum of [0, 0, ..., 0 (n-1 0's), 4, 2*n-2, 0, 0, ..., 0 (n 0's)] and row n of A333275.

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows)

Lars Blomberg, Pdf printout of Excel spreadsheet showing first 100 rows

Scott R. Shannon, Colored illustration for n=1

Scott R. Shannon, Colored illustration for n=2

Scott R. Shannon, Colored illustration for n=3

Scott R. Shannon, Colored illustration for n=4

Scott R. Shannon, Colored illustration for n=5

Scott R. Shannon, Colored illustration for n=6

Scott R. Shannon, Image of the vertices for n = 3.

Scott R. Shannon, Image of the vertices for n = 5.

Scott R. Shannon, Image of the vertices for n = 8.

Scott R. Shannon, Image of the vertices for n = 10.

Scott R. Shannon, Image of the vertices for n = 14.

EXAMPLE

Led d denote the number of polygons meeting at a vertex (except for boundary points, d is the degree of the vertex).

For n=2, the 4 corners have d=3, and on the center line there are 2 vertices with d=4 and 1 with d=6. In the interiors of each of the two squares there are 3 points with d=4.

So in total there are 4 points with d=3, 8 with d=4, and 1 with d=6. So row 2 of the triangle is [0, 4, 8, 0, 1].

The triangle begins:

4,0,1,

0,4,8,0,1,

0,0,28,4,2,0,1,

0,0,54,4,14,0,2,0,1,

0,0,124,0,22,8,2,0,2,0,1,

0,0,214,0,32,4,20,0,2,0,2,0,1;

0,0,382,0,50,0,26,12,2,0,2,0,2,0,1;

0,0,598,0,102,0,18,4,26,0,2,0,2,0,2,0,1;

0,0,950,0,126,0,32,0,30,16,2,0,2,0,2,0,2,0,1;

0,0,1334,0,198,0,62,0,20,4,32,0,2,0,2,0,2,0,2,0,1;

0,0,1912,0,286,0,100,0,10,0,34,20,2,0,2,0,2,0,2,0,2,0,1;

0,0,2622,0,390,0,118,0,38,0,22,4,38,0,2,0,2,0,2,0,2,0,2,0,1;

0,0,3624,0,510,0,136,0,74,0,10,0,38,24,2,0,2,0,2,0,2,0,2,0,2,0,1;

0,0,4690,0,742,0,154,0,118,0,10,0,24,4,44,0,2,0,2,0,2,0,2,0,2,0,2,0,1;

CROSSREFS

KEYWORD

nonn,tabf

AUTHOR

Scott R. Shannon and N. J. A. Sloane, Mar 14 2020

STATUS

approved