A332357 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of cells (both 3-sided and 4-sided) in the partition, for m >= n >= 1.

1, 2, 5, 3, 9, 17, 4, 14, 28, 47, 5, 20, 41, 70, 105, 6, 27, 57, 99, 150, 215, 7, 35, 75, 131, 199, 286, 381, 8, 44, 96, 169, 258, 372, 497, 649, 9, 54, 119, 211, 323, 467, 625, 817, 1029, 10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563, 11, 77, 173, 309, 475, 689, 923, 1208, 1523, 1878, 2257

Offset

1,2

Links

Table of n, a(n) for n=1..66.

M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.

N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Formula

T(m,n) = A332354(m,n)+A332356(m,n).

Example

Triangle begins:
1,
2, 5,
3, 9, 17,
4, 14, 28, 47,
5, 20, 41, 70, 105,
6, 27, 57, 99, 150, 215,
7, 35, 75, 131, 199, 286, 381,
8, 44, 96, 169, 258, 372, 497, 649,
9, 54, 119, 211, 323, 467, 625, 817, 1029,
10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563,
...

Maple

See A332354 and A332356.

Crossrefs

Cf. A332350, A332352, A332354, A332359 (edges).

Main diagonal is A332358.

Sequence in context: A352783 A193972 A361398 * A305126 A044043 A133128

Adjacent sequences: A332354 A332355 A332356 * A332358 A332359 A332360

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 11 2020

Status

approved