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.

4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396

Offset: 2

Author

N. J. A. Sloane, Feb 12 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
4,
8, 20,
12, 32, 52,
16, 48, 80, 124,
20, 64, 108, 168, 228,
24, 84, 144, 228, 312, 428,
28, 104, 180, 288, 396, 544, 692,
32, 128, 224, 360, 496, 684, 872, 1100,
36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
...
		

Links

• M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 12.
• N. J. A. Sloane, Illustration for m=n=3

Crossrefs

Cf. A332350, A332352, A331781, A332371, A332372, A332374.

For main diagonal see A332368.

Programs

Maple

# Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
VS := proc(m,n) local a,i,j; a:=0; # A331781
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
c3 := (m,n) -> VR(m,n,2)+4; # A332367
for m from 2 to 12 do lprint([seq(c3(m,n),n=2..m)]); od:
[seq(c3(n,n)/4,n=2..40)]; # A332368
c4 := (m,n) -> VR(m,n,1)/2 - VR(m,n,2) - 3; # A332369
for m from 2 to 12 do lprint([seq(c4(m,n),n=2..m)]); od:
[seq(c4(n,n),n=2..40)]; # A332370
ct := (m,n) -> c3(m,n)+c4(m,n); # A332371
for m from 2 to 12 do lprint([seq(ct(m,n),n=2..m)]); od:
[seq(ct(n,n),n=2..40)]; # A114043
et := (m,n) -> VR(m,n,1) - VR(m,n,2)/2 - VS(m,n) - 2; # A332372
for m from 2 to 12 do lprint([seq(et(m,n),n=2..m)]); od:
[seq(et(n,n),n=2..40)]; # A332373
vt := (m,n) ->  et(m,n) - ct(m,n) +1; # A332374
for m from 2 to 12 do lprint([seq(vt(m,n),n=2..m)]); od:
[seq(vt(n,n),n=2..40)]; # A332375