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

1,1

Comments

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.

Links

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.

Example

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;
  ...

Crossrefs

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

Sequence in context: A294963 A014198 A316316 * A159786 A083744 A255992

Adjacent sequences: A333285 A333286 A333287 * A333289 A333290 A333291

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 20 2020

Extensions

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

Status

approved