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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. 7
 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 (list; graph; refs; listen; history; text; internal format)
 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. LINKS Lars Blomberg, Table of n, a(n) for n = 1..10200 (the first 100 rows) 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. 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. Sequence in context: A240315 A256041 A137378 * A293071 A084680 A051626 Adjacent sequences:  A333272 A333273 A333274 * A333276 A333277 A333278 KEYWORD nonn,tabf,more 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

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Last modified September 23 12:53 EDT 2020. Contains 337310 sequences. (Running on oeis4.)