cp's OEIS Frontend

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.

%I A332357 #19 Mar 09 2020 22:09:11
%S A332357 1,2,5,3,9,17,4,14,28,47,5,20,41,70,105,6,27,57,99,150,215,7,35,75,
%T A332357 131,199,286,381,8,44,96,169,258,372,497,649,9,54,119,211,323,467,625,
%U A332357 817,1029,10,65,145,258,396,574,769,1006,1268,1563,11,77,173,309,475,689,923,1208,1523,1878,2257
%N 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.
%H A332357 M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.
%H A332357 N. J. A. Sloane, <a href="/A332357/a332357.pdf">Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3)</a> [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
%F A332357 T(m,n) = A332354(m,n)+A332356(m,n).
%e A332357 Triangle begins:
%e A332357 1,
%e A332357 2, 5,
%e A332357 3, 9, 17,
%e A332357 4, 14, 28, 47,
%e A332357 5, 20, 41, 70, 105,
%e A332357 6, 27, 57, 99, 150, 215,
%e A332357 7, 35, 75, 131, 199, 286, 381,
%e A332357 8, 44, 96, 169, 258, 372, 497, 649,
%e A332357 9, 54, 119, 211, 323, 467, 625, 817, 1029,
%e A332357 10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563,
%e A332357 ...
%p A332357 See A332354 and A332356.
%Y A332357 Cf. A332350, A332352, A332354, A332359 (edges).
%Y A332357 Main diagonal is A332358.
%K A332357 nonn,tabl
%O A332357 1,2
%A A332357 _N. J. A. Sloane_, Feb 11 2020