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A332354 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 triangular cells in the partition, for m >= n >= 1.

%I A332354 #17 Feb 12 2020 01:06:20
%S A332354 1,2,5,3,8,15,4,11,22,33,5,14,31,48,71,6,17,40,63,94,125,7,20,51,82,
%T A332354 125,168,227,8,23,62,101,156,211,286,361,9,26,75,124,193,262,357,452,
%U A332354 567,10,29,88,147,230,313,428,543,682,821,11,32,103,174,275,376,517,658,829,1000,1219
%N A332354 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 triangular cells in the partition, for m >= n >= 1.
%H A332354 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 A332354 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]
%e A332354 Triangle begins:
%e A332354 1,
%e A332354 2, 5,
%e A332354 3, 8, 15,
%e A332354 4, 11, 22, 33,
%e A332354 5, 14, 31, 48, 71,
%e A332354 6, 17, 40, 63, 94, 125,
%e A332354 7, 20, 51, 82, 125, 168, 227,
%e A332354 8, 23, 62, 101, 156, 211, 286, 361,
%e A332354 9, 26, 75, 124, 193, 262, 357, 452, 567,
%e A332354 10, 29, 88, 147, 230, 313, 428, 543, 682, 821,
%e A332354 ...
%p A332354 # VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference.
%p A332354 VR := proc(m,n,q) local a,i,j; a:=0;
%p A332354 for i from -m+1 to m-1 do for j from -n+1 to n-1 do
%p A332354 if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
%p A332354 ct3 := proc(m,n) local i; global VR;
%p A332354 if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end;
%p A332354 for m from 1 to 12 do lprint([seq(ct3(m,n),n=1..m)]); od:
%Y A332354 Cf. A332350, A332352, A332356.
%Y A332354 Main diagonal is A332355.
%K A332354 nonn,tabl
%O A332354 1,2
%A A332354 _N. J. A. Sloane_, Feb 11 2020