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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.

Index entries for sequences related to stained glass windows

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.)