OFFSET
1,6
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Feb 11 2020
STATUS
approved