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

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

Author

Felix Huber, Jun 02 2024

Keywords

• nonn

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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);