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

1,6

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.

Equivalently, T(n,2*k) is the number of rational pairs of numbers (x,y) with 0<y<1 such that the equation x-m = y*(p-m) has exactly k integer solutions (m,p) with 0 <= m,p <= n. (x,y) are the coordinates of vertices, and the line goes through the points (m,0) and (p,1). Reinterpreting (m,p) as coordinates, each such pair (x,y) corresponds in the setting of A119437 to a line p = b*m + c with b = -y/(1-y) and c = x/(1-y), which gives all solutions enumerated by that table with negative slope b, and so half the total number of solutions unless horizontal or vertical lines play a role. This explains the formula. See Karlsson. - Andrei Zabolotskii, Jun 23 2026

Links

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

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Jonas Karlsson, Church Windows, Jun 22 2026.

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

Formula

T(n, 2*k) = A119437(n+1, k) / 2 unless k = n+1. - Andrei Zabolotskii, Jun 23 2026

Example

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

Crossrefs

Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694, A119437.

Columns: A334701, A336489, A336490, A336491.

Sequence in context: A339431 A256041 A137378 * A386877 A293071 A084680

Adjacent sequences: A333272 A333273 A333274 * A333276 A333277 A333278

Keyword

nonn,tabf

Author

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

Extensions

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

Status

approved