A333274 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 vertices in the graph at which k polygons meet.

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

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

Cf. A306302, A331755, A331757, A331452, A333275, A333276, A333277.

Sequence in context: A359010 A143784 A272774 * A147311 A147312 A352771

Adjacent sequences: A333271 A333272 A333273 * A333275 A333276 A333277

Keyword

nonn,tabf

Author

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

Status

approved