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%I #53 Jun 21 2020 16:55:54
%S 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,
%T 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,
%U 0,1,0,0,598,0,102,0,18,4,26,0,2,0,2,0,2,0,1
%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 vertices in the graph at which k polygons meet.
%C For vertices not on the boundary, the number of polygons meeting at a vertex is simply the degree (or valency) of that vertex.
%C Row sums are A331755.
%C Sum_k k*T(n,k) gives A333276.
%C See A333275 for the degrees of the non-boundary vertices.
%C 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.
%H Lars Blomberg, <a href="/A333274/b333274.txt">Table of n, a(n) for n = 1..10200</a> (the first 100 rows)
%H Lars Blomberg, <a href="/A333274/a333274.pdf">Pdf printout of Excel spreadsheet showing first 100 rows</a>
%H Scott R. Shannon, <a href="/A331452/a331452_6.png">Colored illustration for n=1</a>
%H Scott R. Shannon, <a href="/A331452/a331452_7.png">Colored illustration for n=2</a>
%H Scott R. Shannon, <a href="/A331452/a331452_8.png">Colored illustration for n=3</a>
%H Scott R. Shannon, <a href="/A331452/a331452_9.png">Colored illustration for n=4</a>
%H Scott R. Shannon, <a href="/A331452/a331452_10.png">Colored illustration for n=5</a>
%H Scott R. Shannon, <a href="/A331452/a331452_11.png">Colored illustration for n=6</a>
%H Scott R. Shannon, <a href="/A333274/a333274_4.png">Image of the vertices for n = 3</a>.
%H Scott R. Shannon, <a href="/A333274/a333274.png">Image of the vertices for n = 5</a>.
%H Scott R. Shannon, <a href="/A333274/a333274_2.png">Image of the vertices for n = 8</a>.
%H Scott R. Shannon, <a href="/A333274/a333274_1.png">Image of the vertices for n = 10</a>.
%H Scott R. Shannon, <a href="/A333274/a333274_3.png">Image of the vertices for n = 14</a>.
%e Led d denote the number of polygons meeting at a vertex (except for boundary points, d is the degree of the vertex).
%e 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.
%e 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].
%e The triangle begins:
%e 4,0,1,
%e 0,4,8,0,1,
%e 0,0,28,4,2,0,1,
%e 0,0,54,4,14,0,2,0,1,
%e 0,0,124,0,22,8,2,0,2,0,1,
%e 0,0,214,0,32,4,20,0,2,0,2,0,1;
%e 0,0,382,0,50,0,26,12,2,0,2,0,2,0,1;
%e 0,0,598,0,102,0,18,4,26,0,2,0,2,0,2,0,1;
%e 0,0,950,0,126,0,32,0,30,16,2,0,2,0,2,0,2,0,1;
%e 0,0,1334,0,198,0,62,0,20,4,32,0,2,0,2,0,2,0,2,0,1;
%e 0,0,1912,0,286,0,100,0,10,0,34,20,2,0,2,0,2,0,2,0,2,0,1;
%e 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;
%e 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;
%e 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;
%Y Cf. A306302, A331755, A331757, A331452, A333275, A333276, A333277.
%K nonn,tabf
%O 1,1
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 14 2020
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