A332356 - OEIS

%I #19 Mar 09 2023 08:37:19

%S 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,

%T 154,0,21,34,68,102,161,211,288,0,28,44,87,130,205,268,365,462,0,36,

%U 57,111,166,261,341,463,586,742,0,45,70,135,200,313,406,550,694,878,1038

%N 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.

%H M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. <a href="https://doi.org/10.1137/140978090">On the minimal teaching sets of two-dimensional threshold functions</a>. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.

%H N. J. A. Sloane, <a href="/A332357/a332357.pdf">Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3)</a> [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

%e Triangle begins:

%e 0,

%e 0, 0,

%e 0, 1, 2,

%e 0, 3, 6, 14,

%e 0, 6, 10, 22, 34,

%e 0, 10, 17, 36, 56, 90,

%e 0, 15, 24, 49, 74, 118, 154,

%e 0, 21, 34, 68, 102, 161, 211, 288,

%e 0, 28, 44, 87, 130, 205, 268, 365, 462,

%e 0, 36, 57, 111, 166, 261, 341, 463, 586, 742,

%e ...

%p # VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference.

%p VR := proc(m,n,q) local a,i,j; a:=0;

%p for i from -m+1 to m-1 do for j from -n+1 to n-1 do

%p if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;

%p ct4 := proc(m,n) local i; global VR;

%p 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;

%p for m from 1 to 12 do lprint([seq(ct4(m,n),n=1..m)]); od:

%t 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];

%t ct4 [m_, n_] := If[m == 1 || n == 1, 0, VR[m, n, 1]/4 - VR[m, n, 2]/2 - m/2 - n/2 - 1];

%t Table[ct4[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* _Jean-François Alcover_, Mar 09 2023, after Maple code *)

%Y Cf. A332350, A332352, A332354.

%Y Main diagonal is A324043.

%K nonn,tabl

%O 1,6

%A _N. J. A. Sloane_, Feb 11 2020