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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.

%I A333275 #34 Jul 09 2025 04:51:19
%S A333275 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 A333275 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 A333275 1,0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
%N 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.
%C A333275 The number of polygons meeting at a non-boundary vertex is simply the degree (or valency) of that vertex.
%C A333275 Row sums are A159065.
%C A333275 Sum_k k*T(n,k) gives A333277.
%C A333275 See A333274 for the degrees if the boundary vertices are included.
%C A333275 T(n,k) = 0 if k is odd. But the triangle includes those zero entries because this is used to construct A333274.
%H A333275 Lars Blomberg, <a href="/A333275/b333275.txt">Table of n, a(n) for n = 1..10200</a> (the first 100 rows)
%H A333275 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 A333275 Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration showing regions for n=1</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755.png">Images of vertices for n=1</a>.
%H A333275 Scott R. Shannon, <a href="/A331452/a331452_7.png">Colored illustration showing regions for n=2</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_1.png">Images of vertices for n=2</a>.
%H A333275 Scott R. Shannon, <a href="/A331452/a331452_8.png">Colored illustration showing regions for n=3</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_2.png">Images of vertices for n=3</a>.
%H A333275 Scott R. Shannon, <a href="/A331452/a331452_9.png">Colored illustration showing regions for n=4</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_3.png">Images of vertices for n=4</a>.
%H A333275 Scott R. Shannon, <a href="/A331452/a331452_10.png">Colored illustration showing regions for n=5</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_7.png">Images of vertices for n=5</a>
%H A333275 Scott R. Shannon, <a href="/A331452/a331452_11.png">Colored illustration showing regions for n=6</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_8.png">Images of vertices for n=6</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_11.png">Images of vertices for n=7</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_10.png">Images of vertices for n=8</a>
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_4.png">Images of vertices for n=9</a>.
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_5.png">Images of vertices for n=11</a>.
%H A333275 Scott R. Shannon, <a href="/A331755/a331755_6.png">Images of vertices for n=14</a>.
%H A333275 <a href="/index/St#Stained">Index entries for sequences related to stained glass windows</a>
%e A333275 Led d denote the number of polygons meeting at a vertex.
%e A333275 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 A333275 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 A333275 The triangle begins:
%e A333275 0,0,1,
%e A333275 0,0,6,0,1,
%e A333275 0,0,24,0,2,0,1,
%e A333275 0,0,54,0,8,0,2,0,1,
%e A333275 0,0,124,0,18,0,2,0,2,0,1,
%e A333275 0,0,214,0,32,0,10,0,2,0,2,0,1,
%e A333275 0,0,382,0,50,0,22,0,2,0,2,0,2,0,1,
%e A333275 0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1
%e A333275 ...
%e A333275 If we leave out the uninteresting zeros, the triangle begins:
%e A333275 [1]
%e A333275 [6, 1]
%e A333275 [24, 2, 1]
%e A333275 [54, 8, 2, 1]
%e A333275 [124, 18, 2, 2, 1]
%e A333275 [214, 32, 10, 2, 2, 1]
%e A333275 [382, 50, 22, 2, 2, 2, 1]
%e A333275 [598, 102, 18, 12, 2, 2, 2, 1]
%e A333275 [950, 126, 32, 26, 2, 2, 2, 2, 1]
%e A333275 [1334, 198, 62, 20, 14, 2, 2, 2, 2, 1]
%e A333275 [1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1]
%e A333275 [2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1]
%e A333275 ... - _N. J. A. Sloane_, Jul 27 2020
%Y A333275 Cf. A306302, A331755, A331757, A331452, A333274, A333276, A333277, A334694.
%K A333275 nonn,tabf
%O A333275 1,6
%A A333275 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 14 2020
%E A333275 a(36) and beyond from _Lars Blomberg_, Jun 17 2020