A335056 -id:A335056 - cp's OEIS frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

4, 8, 8, 12, 22, 12, 16, 36, 36, 16, 20, 52, 70, 52, 20, 24, 66, 100, 100, 66, 24, 28, 82, 134, 160, 134, 82, 28, 32, 98, 166, 218, 218, 166, 98, 32, 36, 116, 198, 276, 310, 276, 198, 116, 36, 40, 134, 230, 328, 396, 396, 328, 230, 134, 40, 44, 154, 266, 386

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.
Since the cells are either triangles or quadrilaterals, this is the sum of the two arrays A333286 and A333287.
It would be nice to have a formula for these entries. It is easy to see that the first column is 4n for n>=1.

			Triangle begins:
   4;
   8,   8;
  12,  22,  12;
  16,  36,  36,  16;
  20,  52,  70,  52,  20;
  24,  66, 100, 100,  66,  24;
  28,  82, 134, 160, 134,  82,  28;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333287, A333288, A335056, A335074.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A335074 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all possible rectangles; T(n,k) (1 <= k <= n-1) is the number of vertices on the edge separating rectangles k and k+1.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 7, 7, 7, 9, 9, 7, 9, 9, 11, 13, 11, 11, 13, 11, 13, 15, 17, 11, 17, 15, 13, 15, 19, 19, 19, 19, 19, 19, 15, 17, 21, 25, 21, 19, 21, 25, 21, 17, 19, 25, 29, 29, 23, 23, 29, 29, 25, 19, 21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21, 23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23

Offset: 2

Scott R. Shannon and N. J. A. Sloane, May 22 2020

The terms are from numeric computation - no formula for a(n) is currently known.

			Triangle begins:
1;
3, 3;
5, 3, 5;
7, 7, 7, 7;
9, 9, 7, 9, 9;
11, 13, 11, 11, 13, 11;
13, 15, 17, 11, 17, 15, 13;
15, 19, 19, 19, 19, 19, 19, 15;
17, 21, 25, 21, 19, 21, 25, 21, 17;
19, 25, 29, 29, 23, 23, 29, 29, 25, 19;
21, 27, 33, 33, 33, 23, 33, 33, 33, 27, 21;
23, 31, 37, 39, 39, 35, 35, 39, 39, 37, 31, 23;

Scott R. Shannon, Image for n = 3 showing the count of the vertices.
Scott R. Shannon, Image for n = 4 showing the count of the vertices.
Scott R. Shannon, Image for n = 7 showing the count of the vertices.
Scott R. Shannon, Image for n = 10 showing the count of the vertices.
Scott R. Shannon, Image for n = 12 showing the count of the vertices.

Cf. A335056, A159065, A331755, A333288, A306302.

Row sum n + Row sum A335056(n) = A159065(n).

cp's OEIS Frontend

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

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