%I #32 Jan 01 2024 22:47:07

%S 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,

%T 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,

%U 1,0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1

%N 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.

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

%C Row sums are A159065.

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

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

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

%H Lars Blomberg, <a href="/A333275/b333275.txt">Table of n, a(n) for n = 1..10200</a> (the first 100 rows)

%H Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, <a href="http://neilsloane.com/doc/rose_5.pdf">Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids</a>, (2021). Also arXiv:2009.07918.

%H Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration showing regions for n=1</a>

%H Scott R. Shannon, <a href="/A331755/a331755.png">Images of vertices for n=1</a>.

%H Scott R. Shannon, <a href="/A331452/a331452_7.png">Colored illustration showing regions for n=2</a>

%H Scott R. Shannon, <a href="/A331755/a331755_1.png">Images of vertices for n=2</a>.

%H Scott R. Shannon, <a href="/A331452/a331452_8.png">Colored illustration showing regions for n=3</a>

%H Scott R. Shannon, <a href="/A331755/a331755_2.png">Images of vertices for n=3</a>.

%H Scott R. Shannon, <a href="/A331452/a331452_9.png">Colored illustration showing regions for n=4</a>

%H Scott R. Shannon, <a href="/A331755/a331755_3.png">Images of vertices for n=4</a>.

%H Scott R. Shannon, <a href="/A331452/a331452_10.png">Colored illustration showing regions for n=5</a>

%H Scott R. Shannon, <a href="/A331755/a331755_7.png">Images of vertices for n=5</a>

%H Scott R. Shannon, <a href="/A331452/a331452_11.png">Colored illustration showing regions for n=6</a>

%H Scott R. Shannon, <a href="/A331755/a331755_8.png">Images of vertices for n=6</a>

%H Scott R. Shannon, <a href="/A331755/a331755_11.png">Images of vertices for n=7</a>

%H Scott R. Shannon, <a href="/A331755/a331755_10.png">Images of vertices for n=8</a>

%H Scott R. Shannon, <a href="/A331755/a331755_4.png">Images of vertices for n=9</a>.

%H Scott R. Shannon, <a href="/A331755/a331755_5.png">Images of vertices for n=11</a>.

%H Scott R. Shannon, <a href="/A331755/a331755_6.png">Images of vertices for n=14</a>.

%H <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>

%e Led d denote the number of polygons meeting at a vertex.

%e 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.

%e 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].

%e The triangle begins:

%e 0,0,1,

%e 0,0,6,0,1,

%e 0,0,24,0,2,0,1,

%e 0,0,54,0,8,0,2,0,1,

%e 0,0,124,0,18,0,2,0,2,0,1,

%e 0,0,214,0,32,0,10,0,2,0,2,0,1,

%e 0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,

%e 0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1

%e ...

%e If we leave out the uninteresting zeros, the triangle begins:

%e [1]

%e [6, 1]

%e [24, 2, 1]

%e [54, 8, 2, 1]

%e [124, 18, 2, 2, 1]

%e [214, 32, 10, 2, 2, 1]

%e [382, 50, 22, 2, 2, 2, 1]

%e [598, 102, 18, 12, 2, 2, 2, 1]

%e [950, 126, 32, 26, 2, 2, 2, 2, 1]

%e [1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]

%e [1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]

%e [2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]

%e ... - _N. J. A. Sloane_, Jul 27 2020

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

%K nonn,tabf

%O 1,6

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 14 2020.

%E a(36) and beyond from _Lars Blomberg_, Jun 17 2020