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A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

%I A332367 #22 Apr 27 2020 09:24:06
%S A332367 4,8,20,12,32,52,16,48,80,124,20,64,108,168,228,24,84,144,228,312,428,
%T A332367 28,104,180,288,396,544,692,32,128,224,360,496,684,872,1100,36,152,
%U A332367 268,432,596,824,1052,1328,1604,40,180,320,520,720,1000,1280,1620,1960,2396
%N A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.
%H A332367 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 12.
%H A332367 N. J. A. Sloane, <a href="/A332371/a332371.pdf">Illustration for m=n=3</a>
%e A332367 Triangle begins:
%e A332367 4,
%e A332367 8, 20,
%e A332367 12, 32, 52,
%e A332367 16, 48, 80, 124,
%e A332367 20, 64, 108, 168, 228,
%e A332367 24, 84, 144, 228, 312, 428,
%e A332367 28, 104, 180, 288, 396, 544, 692,
%e A332367 32, 128, 224, 360, 496, 684, 872, 1100,
%e A332367 36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
%e A332367 ...
%p A332367 # Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
%p A332367 VR := proc(m,n,q) local a,i,j; a:=0;
%p A332367 for i from -m+1 to m-1 do for j from -n+1 to n-1 do
%p A332367 if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
%p A332367 VS := proc(m,n) local a,i,j; a:=0; # A331781
%p A332367 for i from 1 to m-1 do for j from 1 to n-1 do
%p A332367 if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
%p A332367 c3 := (m,n) -> VR(m,n,2)+4; # A332367
%p A332367 for m from 2 to 12 do lprint([seq(c3(m,n),n=2..m)]); od:
%p A332367 [seq(c3(n,n)/4,n=2..40)]; # A332368
%p A332367 c4 := (m,n) -> VR(m,n,1)/2 - VR(m,n,2) - 3; # A332369
%p A332367 for m from 2 to 12 do lprint([seq(c4(m,n),n=2..m)]); od:
%p A332367 [seq(c4(n,n),n=2..40)]; # A332370
%p A332367 ct := (m,n) -> c3(m,n)+c4(m,n); # A332371
%p A332367 for m from 2 to 12 do lprint([seq(ct(m,n),n=2..m)]); od:
%p A332367 [seq(ct(n,n),n=2..40)]; # A114043
%p A332367 et := (m,n) -> VR(m,n,1) - VR(m,n,2)/2 - VS(m,n) - 2; # A332372
%p A332367 for m from 2 to 12 do lprint([seq(et(m,n),n=2..m)]); od:
%p A332367 [seq(et(n,n),n=2..40)]; # A332373
%p A332367 vt := (m,n) ->  et(m,n) - ct(m,n) +1; # A332374
%p A332367 for m from 2 to 12 do lprint([seq(vt(m,n),n=2..m)]); od:
%p A332367 [seq(vt(n,n),n=2..40)]; # A332375
%Y A332367 Cf. A332350, A332352, A331781, A332371, A332372, A332374.
%Y A332367 For main diagonal see A332368.
%K A332367 nonn,tabl
%O A332367 2,1
%A A332367 _N. J. A. Sloane_, Feb 12 2020