A334713 -id:A334713 - cp's OEIS frontend

A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 32, 48, 32, 0, 0, 100, 168, 168, 100, 0, 0, 240, 456, 532, 456, 240, 0, 0, 490, 990, 1312, 1312, 990, 490, 0, 0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0, 0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0, 0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, ...
0, 6, 48, 168, 456, 990, 1920, 3360, 5520, 8550, 12720, 18216, ...
0, 32, 168, 532, 1312, 2652, 4972, 8420, 13452, 20480, 29980, 42288, ...
0, 100, 456, 1312, 3088, 5964, 10816, 17768, 27840, 41652, 60040, 83448, ...
0, 240, 990, 2652, 5964, 11340, 20142, 32436, 50004, 73704, 105282, 144936, ...
0, 490, 1920, 4972, 10816, 20142, 35264, 55916, 84960, 123690, 174976, 238512, ...
0, 896, 3360, 8420, 17768, 32436, 55916, 88088, 132708, 191588, 268972, 363876, ...
0, 1512, 5520, 13452, 27840, 50004, 84960, 132708, 198912, 285312, 397968, 534888, ...
0, 2400, 8550, 20480, 41652, 73704, 123690, 191588, 285312, 407744, 566046, 757008, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
0, 6, 6, 0
0, 32, 48, 32, 0
0, 100, 168, 168, 100, 0
0, 240, 456, 532, 456, 240, 0
0, 490, 990, 1312, 1312, 990, 490, 0
0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0
0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0
0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0
0, 3630, 8550, 13452, 17768, 20142, 20142, 17768, 13452, 8550, 3630, 0
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A334713.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A372915 a(n) is the number of distinct triangles with area n whose vertices are points of an n X n grid.

Original entry on oeis.org

0, 0, 2, 4, 9, 10, 25, 22, 38, 49, 56, 56, 111, 71, 119, 141, 153, 126, 249, 166, 244, 299, 279, 244, 463, 288, 361, 489, 517, 373, 677, 436, 626, 719, 620, 665, 1078, 604, 811, 936, 1000, 749, 1444, 842, 1221, 1384, 1173, 1016, 1871, 1261, 1393, 1597, 1566, 1259

Offset: 0

Felix Huber, Jun 02 2024

nonn

			See the linked illustration for the term a(4) = 9.

Felix Huber, Table of n, a(n) for n=0..666
Felix Huber, Illustration of the term a(4) = 9

Cf. A045996, A088658, A303331, A320310, A320542, A320544, A334713, A372217, A372218.

A372915:=proc(n)
  local p,q,g,h,u,v,x,y,L,M;
  L:=[];
  for g from 2 to n do
    h:=2*n/g;
    if type(h,integer) then
      for x to n do
        M:=[g,sqrt(x^2+h^2),sqrt((g-x)^2+h^2)];
        M:=sort(M);
        if not member(M,L) then
          L:=[op(L),M];
        fi;
      od;
    fi;
  od;
  for p to n do
    for q from 1 to p do
      g:=sqrt(p^2+q^2);
      h:=2*n/g;
      u:=h/g*q;
      v:=q+h/g*p;
      for x from max(1,ceil(p/q*(v-n)+u)) to min(n,floor(p/q*v+u)) do
        y:=q/p*(u-x)+v;
        if type(y,integer) and x <> p and y <> q then
          M:=[g,sqrt(x^2+(y-q)^2),sqrt((x-p)^2+y^2)];
          M:=sort(M);
          if not member(M,L) then
            L:=[op(L),M];
          fi;
        fi;
      od;
    od;
  od;
  return numelems(L);
end proc;
seq(A372915(n),n=0..53);

cp's OEIS Frontend

A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

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