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A332356 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 quadrilateral cells in the partition, for m >= n >= 1.

%I A332356 #19 Mar 09 2023 08:37:19
%S A332356 0,0,0,0,1,2,0,3,6,14,0,6,10,22,34,0,10,17,36,56,90,0,15,24,49,74,118,
%T A332356 154,0,21,34,68,102,161,211,288,0,28,44,87,130,205,268,365,462,0,36,
%U A332356 57,111,166,261,341,463,586,742,0,45,70,135,200,313,406,550,694,878,1038
%N A332356 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 quadrilateral cells in the partition, for m >= n >= 1.
%H A332356 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 A332356 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 A332356 Triangle begins:
%e A332356 0,
%e A332356 0, 0,
%e A332356 0, 1, 2,
%e A332356 0, 3, 6, 14,
%e A332356 0, 6, 10, 22, 34,
%e A332356 0, 10, 17, 36, 56, 90,
%e A332356 0, 15, 24, 49, 74, 118, 154,
%e A332356 0, 21, 34, 68, 102, 161, 211, 288,
%e A332356 0, 28, 44, 87, 130, 205, 268, 365, 462,
%e A332356 0, 36, 57, 111, 166, 261, 341, 463, 586, 742,
%e A332356 ...
%p A332356 # VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference.
%p A332356 VR := proc(m,n,q) local a,i,j; a:=0;
%p A332356 for i from -m+1 to m-1 do for j from -n+1 to n-1 do
%p A332356 if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
%p A332356 ct4 := proc(m,n) local i; global VR;
%p A332356 if m=1 or n=1 then 0 else VR(m,n,1)/4-VR(m,n,2)/2-m/2-n/2-1; fi; end;
%p A332356 for m from 1 to 12 do lprint([seq(ct4(m,n),n=1..m)]); od:
%t A332356 VR[m_, n_, q_] := Module[{a = 0, i, j}, For[i = -m + 1, i <= m - 1, i++, For[j = -n + 1, j <= n - 1, j++, If[GCD[i, j] == q, a = a + (m - Abs[i])*(n - Abs[j])]]];a];
%t A332356 ct4 [m_, n_] := If[m == 1 || n == 1, 0, VR[m, n, 1]/4 - VR[m, n, 2]/2 - m/2 - n/2 - 1];
%t A332356 Table[ct4[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Mar 09 2023, after Maple code *)
%Y A332356 Cf. A332350, A332352, A332354.
%Y A332356 Main diagonal is A324043.
%K A332356 nonn,tabl
%O A332356 1,6
%A A332356 _N. J. A. Sloane_, Feb 11 2020