A372218 -id:A372218 - cp's OEIS frontend

A372217 a(n) is the number of distinct triangles whose sides do not pass through a grid point and whose vertices are three points of an n X n grid.

0, 1, 3, 8, 14, 36, 48, 100, 146, 232, 294, 502, 595, 938, 1143, 1433, 1741, 2512, 2826, 3911, 4458, 5319, 6067, 7976, 8728, 10750, 12076, 14194, 15671, 19510, 20669, 25349, 28115, 31716, 34697, 39467, 41894, 49766, 54046, 59948, 63951, 74818, 78216, 90773, 97220

Offset: 0

Felix Huber, Apr 28 2024

nonn

			See the linked illustration for the terms a(1) = 1, a(2) = 3, a(3) = 8, a(4) = 14, a(5) = 36 and a(6) = 48.

Felix Huber, Illustration of the terms a(1) to a(6).

Cf. A115004, A141224, A141255, A320540, A320541, A320544, A372218.

S372217:=proc(n);
  local s,x,u,v;
  s:=0;
  if n=1 then return 1 fi;
  for x to n do
    if gcd(x,n)=1 then
      for u from x to n do
        for v from 0 to n do
          if gcd(u,v)=1 and gcd(u-x,n-v)=1 then
            if u=x then s:=s+1;
            fi;
          fi;
        od;
      od;
    fi;
  od;
  return s;
end proc;
A372217:=proc(n)
  local i,a;
  a:=0;
  for i from 0 to n do
    a:=a+S372217(i);
  od;
  return a;
end proc;
seq(A372217(n),n=0..44);

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

A372217 a(n) is the number of distinct triangles whose sides do not pass through a grid point and whose vertices are three points of an n X n grid.

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