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

2,1

Links

Table of n, a(n) for n=2..56.

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

Example

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,
...

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

Crossrefs

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

For main diagonal see A332368.

Sequence in context: A340948 A265108 A328184 * A273143 A273174 A178447

Adjacent sequences: A332364 A332365 A332366 * A332368 A332369 A332370

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 12 2020

Status

approved