A332359 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 edges in the partition, for m >= n >= 1.

3, 5, 10, 7, 17, 30, 9, 26, 49, 82, 11, 37, 71, 121, 180, 13, 50, 99, 172, 259, 374, 15, 65, 130, 227, 342, 495, 656, 17, 82, 167, 294, 445, 646, 859, 1126, 19, 101, 207, 367, 557, 811, 1080, 1417, 1784, 21, 122, 253, 450, 685, 1000, 1333, 1750, 2205, 2726, 23, 145, 302, 539, 821, 1199, 1597, 2097, 2642, 3267, 3916

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) = (3*A332354(m,n) + 4*A332356(m,n) + m + n + 1)/2.

Example

Triangle begins:
3,
5, 10,
7, 17, 30,
9, 26, 49, 82,
11, 37, 71, 121, 180,
13, 50, 99, 172, 259, 374,
15, 65, 130, 227, 342, 495, 656,
17, 82, 167, 294, 445, 646, 859, 1126,
19, 101, 207, 367, 557, 811, 1080, 1417, 1784,
21, 122, 253, 450, 685, 1000, 1333, 1750, 2205, 2726,
...

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;
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;
for m from 1 to 12 do lprint([seq(cte(m, n), n=1..m)]); od:

Crossrefs

Cf. A332350, A332352, A332354, A332357 (edges).

Main diagonal is A332360.

Sequence in context: A211414 A173706 A365440 * A328070 A286592 A176629

Adjacent sequences: A332356 A332357 A332358 * A332360 A332361 A332362

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 11 2020

Status

approved