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A332359 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 edges in the partition, for m >= n >= 1.

%I A332359 #14 Feb 12 2020 01:05:54
%S A332359 3,5,10,7,17,30,9,26,49,82,11,37,71,121,180,13,50,99,172,259,374,15,
%T A332359 65,130,227,342,495,656,17,82,167,294,445,646,859,1126,19,101,207,367,
%U A332359 557,811,1080,1417,1784,21,122,253,450,685,1000,1333,1750,2205,2726,23,145,302,539,821,1199,1597,2097,2642,3267,3916
%N A332359 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 edges in the partition, for m >= n >= 1.
%H A332359 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 A332359 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 A332359 T(m,n) = (3*A332354(m,n) + 4*A332356(m,n) + m + n + 1)/2.
%e A332359 Triangle begins:
%e A332359 3,
%e A332359 5, 10,
%e A332359 7, 17, 30,
%e A332359 9, 26, 49, 82,
%e A332359 11, 37, 71, 121, 180,
%e A332359 13, 50, 99, 172, 259, 374,
%e A332359 15, 65, 130, 227, 342, 495, 656,
%e A332359 17, 82, 167, 294, 445, 646, 859, 1126,
%e A332359 19, 101, 207, 367, 557, 811, 1080, 1417, 1784,
%e A332359 21, 122, 253, 450, 685, 1000, 1333, 1750, 2205, 2726,
%e A332359 ...
%p A332359 VR := proc(m,n,q) local a,i,j; a:=0;
%p A332359 for i from -m+1 to m-1 do for j from -n+1 to n-1 do
%p A332359 if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
%p A332359 cte := proc(m,n) local i; global VR;
%p A332359 if m=1 or n=1 then 2*max(m,n)+1 else VR(m,n,1)/2-VR(m,n,2)/4+m+n; fi; end;
%p A332359 for m from 1 to 12 do lprint([seq(cte(m,n),n=1..m)]); od:
%Y A332359 Cf. A332350, A332352, A332354, A332357 (edges).
%Y A332359 Main diagonal is A332360.
%K A332359 nonn,tabl
%O A332359 1,1
%A A332359 _N. J. A. Sloane_, Feb 11 2020