A333287 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 quadrilateral regions in the k-th rectangle.

0, 1, 1, 3, 8, 3, 5, 12, 12, 5, 7, 22, 32, 22, 7, 9, 28, 40, 40, 28, 9, 11, 38, 58, 74, 58, 38, 11, 13, 46, 74, 98, 98, 74, 46, 13, 15, 58, 92, 130, 152, 130, 92, 58, 15, 17, 68, 104, 150, 180, 180, 150, 104, 68, 17, 19, 82, 124, 180, 224, 254, 224, 180, 124, 82, 19

Offset

1,4

Comments

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.

It would be nice to have a formula for these entries. It is easy to see that the first column is 2n-3 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:
0,
1,  1,
3,  8,  3,
5, 12, 12,  5,
7, 22, 32, 22,  7,
9, 28, 40, 40, 28,  9,
11, 38, 58, 74, 58, 38, 11,
...

Crossrefs

Cf. A306302, A331452, A324042, A324043, A333286, A333288.

Sequence in context: A334503 A015136 A179450 * A360199 A360851 A117240

Adjacent sequences: A333284 A333285 A333286 * A333288 A333289 A333290

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 20 2020

Extensions

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

Status

approved