A333275 Irregular triangle read by rows: consider the graph defined in A306302 formed from a row of n adjacent congruent rectangles by drawing the diagonals of all visible rectangles; T(n,k) (n >= 1, 2 <= k <= 2n+2) is the number of non-boundary vertices in the graph at which k polygons meet.

0, 0, 1, 0, 0, 6, 0, 1, 0, 0, 24, 0, 2, 0, 1, 0, 0, 54, 0, 8, 0, 2, 0, 1, 0, 0, 124, 0, 18, 0, 2, 0, 2, 0, 1, 0, 0, 214, 0, 32, 0, 10, 0, 2, 0, 2, 0, 1, 0, 0, 382, 0, 50, 0, 22, 0, 2, 0, 2, 0, 2, 0, 1, 0, 0, 598, 0, 102, 0, 18, 0, 12, 0, 2, 0, 2, 0, 2, 0, 1

Offset: 1

Author

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

Keywords

• nonn
• tabf

Comments

The number of polygons meeting at a non-boundary vertex is simply the degree (or valency) of that vertex.

Row sums are A159065.

Sum_k k*T(n,k) gives A333277.

See A333274 for the degrees if the boundary vertices are included.

T(n,k) = 0 if k is odd. But the triangle includes those zero entries because this is used to construct A333274.

Examples

			Led d denote the number of polygons meeting at a vertex.
For n=2, in the interiors of each of the two squares there are 3 points with d=4, and the center point has d=6.
So in total there are 6 points with d=4 and 1 with d=6. So row 2 of the triangle is [0, 0, 6, 0, 1].
The triangle begins:
0,0,1,
0,0,6,0,1,
0,0,24,0,2,0,1,
0,0,54,0,8,0,2,0,1,
0,0,124,0,18,0,2,0,2,0,1,
0,0,214,0,32,0,10,0,2,0,2,0,1,
0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,
0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
...
If we leave out the uninteresting zeros, the triangle begins:
[1]
[6, 1]
[24, 2, 1]
[54, 8, 2, 1]
[124, 18, 2, 2, 1]
[214, 32, 10, 2, 2, 1]
[382, 50, 22, 2, 2, 2, 1]
[598, 102, 18, 12, 2, 2, 2, 1]
[950, 126, 32, 26, 2, 2, 2, 2, 1]
[1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]
[1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]
[2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]
... - _N. J. A. Sloane_, Jul 27 2020
		

Links

• Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows)
• Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
• Scott R. Shannon, Colored illustration showing regions for n=1
• Scott R. Shannon, Images of vertices for n=1.
• Scott R. Shannon, Colored illustration showing regions for n=2
• Scott R. Shannon, Images of vertices for n=2.
• Scott R. Shannon, Colored illustration showing regions for n=3
• Scott R. Shannon, Images of vertices for n=3.
• Scott R. Shannon, Colored illustration showing regions for n=4
• Scott R. Shannon, Images of vertices for n=4.
• Scott R. Shannon, Colored illustration showing regions for n=5
• Scott R. Shannon, Images of vertices for n=5
• Scott R. Shannon, Colored illustration showing regions for n=6
• Scott R. Shannon, Images of vertices for n=6
• Scott R. Shannon, Images of vertices for n=7
• Scott R. Shannon, Images of vertices for n=8
• Scott R. Shannon, Images of vertices for n=9.
• Scott R. Shannon, Images of vertices for n=11.
• Scott R. Shannon, Images of vertices for n=14.
• Index entries for sequences related to stained glass windows

Crossrefs

Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694.

Extensions

a(36) and beyond from Lars Blomberg, Jun 17 2020