A332356 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 quadrilateral cells in the partition, for m >= n >= 1.

0, 0, 0, 0, 1, 2, 0, 3, 6, 14, 0, 6, 10, 22, 34, 0, 10, 17, 36, 56, 90, 0, 15, 24, 49, 74, 118, 154, 0, 21, 34, 68, 102, 161, 211, 288, 0, 28, 44, 87, 130, 205, 268, 365, 462, 0, 36, 57, 111, 166, 261, 341, 463, 586, 742, 0, 45, 70, 135, 200, 313, 406, 550, 694, 878, 1038

Offset

1,6

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]

Example

Triangle begins:
0,
0, 0,
0, 1, 2,
0, 3, 6, 14,
0, 6, 10, 22, 34,
0, 10, 17, 36, 56, 90,
0, 15, 24, 49, 74, 118, 154,
0, 21, 34, 68, 102, 161, 211, 288,
0, 28, 44, 87, 130, 205, 268, 365, 462,
0, 36, 57, 111, 166, 261, 341, 463, 586, 742,
...

Maple

# VR(m, n, q) is f_q(m, n) from the Alekseyev et al. reference.
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;
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;
for m from 1 to 12 do lprint([seq(ct4(m, n), n=1..m)]); od:

Mathematica

VR[m_, n_, q_] := Module[{a = 0, i, j}, For[i = -m + 1, i <= m - 1, i++, For[j = -n + 1, j <= n - 1, j++, If[GCD[i, j] == q, a = a + (m - Abs[i])*(n - Abs[j])]]]; a];
ct4 [m_, n_] := If[m == 1 || n == 1, 0, VR[m, n, 1]/4 - VR[m, n, 2]/2 - m/2 - n/2 - 1];
Table[ct4[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 09 2023, after Maple code *)

Crossrefs

Cf. A332350, A332352, A332354.

Main diagonal is A324043.

Sequence in context: A330674 A248966 A021495 * A309655 A260591 A127160

Adjacent sequences: A332353 A332354 A332355 * A332357 A332358 A332359

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 11 2020

Status

approved