A331454 Triangle read by rows: T(n,m) (n >= m >= 1) = number of line segments formed by drawing the lines connecting any two of the 2*(m+n) perimeter points of an m X n grid of squares.

8, 28, 92, 80, 240, 596, 178, 508, 1028, 1936, 372, 944, 2004, 3404, 6020, 654, 1548, 3018, 4962, 8064, 11088, 1124, 2520, 4808, 7734, 12708, 17022, 26260, 1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144, 2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296, 3914, 7528, 12720, 19428, 29914, 37848, 54612, 67590, 86906, 107832

Offset

1,1

Comments

Take a grid of m+1 X n+1 points. There are 2*(m+n) points on the perimeter. Join every pair of the perimeter points by a line (of finite length). The lines do not extend outside the grid. T(m,n) is the number of line segments formed when these lines intersect each other, and A331452(m,n) and A331453(m,n) give the number of regions and the number of vertices respectively.

For illustrations see the links in A331452.

Links

Lars Blomberg, Table of n, a(n) for n = 1..703 (the first 37 rows)

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.

Example

Triangle begins:
8,
28, 92,
80, 240, 596,
178, 508, 1028, 1936,
372, 944, 2004, 3404, 6020,
654, 1548, 3018, 4962, 8064, 11088,
1124, 2520, 4808, 7734, 12708, 17022, 26260,
1782, 3754, 6704, 10840, 16608, 22220, 32794, 42144,
2724, 5392, 9780, 14620, 22788, 30238, 44028, 54024, 72296,
...

Crossrefs

The main diagonal is A331448.

The first two columns are A331757 and A331765.

Cf. A288187, A331452, A331453.

Sequence in context: A005343 A200941 A332600 * A333278 A333283 A211066

Adjacent sequences: A331451 A331452 A331453 * A331455 A331456 A331457

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Jan 27 2020

Status

approved