A332361 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 vertices in the partition, for m >= n >= 1.

3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, 13, 69, 130, 231, 347, 511, 675, 890, 1120, 1390, 1660

Offset

1,1

Links

Table of n, a(n) for n=1..66.

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 13.

N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Formula

T(m,n) = A332359(m,n) - A332357(m,n) + 1 (Euler's formula).

Example

Triangle begins:
3,
4, 6,
5, 9, 14,
6, 13, 22, 36,
7, 18, 31, 52, 76,
8, 24, 43, 74, 110, 160,
9, 31, 56, 97, 144, 210, 276,
10, 39, 72, 126, 188, 275, 363, 478,
11, 48, 89, 157, 235, 345, 456, 601, 756,
12, 58, 109, 193, 290, 427, 565, 745, 938, 1164,
...

Maple

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;
ct3 := proc(m, n) local i; global VR;
if m=1 or n=1 then max(m, n) else VR(m, n, 2)/2+m+n+1; fi; end; # A332354
ct4 := proc(m, n) local i; global VR;
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; # A332356
ct := (m, n) -> ct3(m, n) + ct4(m, n); # A332357
cte := proc(m, n) local i; global VR;
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; # A332359
ctv := (m, n) -> cte(m, n) - ct(m, n) + 1; # A332361
for m from 1 to 12 do lprint([seq(ctv(m, n), n=1..m)]); od:

Crossrefs

Cf. A332350-A331360.

Main diagonal is A332362.

Sequence in context: A016655 A057757 A336717 * A360655 A058838 A349372

Adjacent sequences: A332358 A332359 A332360 * A332362 A332363 A332364

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 11 2020

Status

approved