A331766 Number of regions formed by drawing the lines connecting any two of the 2*(n+2) perimeter points of a 3 X (n+1) rectangular grid of points (or equally, a 2 X n grid of squares).

16, 56, 142, 296, 544, 892, 1436, 2136, 3066, 4272, 5840, 7688, 10094, 12884, 16182, 20192, 24918, 30200, 36614, 43692, 51756, 61008, 71544, 83040, 96202, 110692, 126702, 144372, 164144, 185200, 209192, 234928, 262706, 293244, 326002, 361240, 400170, 441516

Offset

1,1

Comments

The grid consists of a rectangular array of 3 X (n+1) dots. If we instead count squares, the dimensions are 2 X n.

Triangles A331452, A331453, A331454 do not have formulas, except for column 1. The column 2 sequences, A331763, A331765, A331766, are therefore the next ones to attack.

See A331452 for other illustrations.

For n<=100, 7-gons: 4 for n=9, 4 for n=18; 8-gons: 2 for n=9; no 9-gons or 10-gons. Lars Blomberg, Apr 28 2020

Links

Lars Blomberg, Table of n, a(n) for n = 1..100

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

Scott R. Shannon, Colored illustration for a(3) = 142.

Scott R. Shannon, Data specifically for nX2 (or 2Xn) rectangles

N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.

Crossrefs

Column 2 of A331452.

Cf. A331763, A331765.

Sequence in context: A347253 A039451 A279063 * A231971 A333279 A304692

Adjacent sequences: A331763 A331764 A331765 * A331767 A331768 A331769

Keyword

nonn

Author

Scott R. Shannon and N. J. A. Sloane, Feb 05 2020

Extensions

More terms from Scott R. Shannon, Mar 11 2020

a(21) and beyond from Lars Blomberg, Apr 28 2020

Status

approved