A190313 Number of scalene triangles, distinct up to congruence, on an n X n grid (or geoboard).

0, 0, 3, 18, 57, 137, 280, 517, 863, 1368, 2069, 3007, 4218, 5774, 7704, 10109, 13025, 16523, 20671, 25567, 31274, 37891, 45529, 54213, 64082, 75320, 87901, 102014, 117736, 135217, 154606, 176024, 199502, 225290, 253485, 284305, 317811, 354282, 393618, 436202, 482332

Offset: 1

Author

Martin Renner, May 08 2011

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Geoboard.
• Eric Weisstein's World of Mathematics, Scalene Triangle.

Crossrefs

Cf. A028419, A189978.

Programs

Mathematica

q[n_] :=
  Module[{sqDist, t0, t1, t2, t3},
   (*Squared distances*)
   sqDist = {p_,q_} :> (Floor[p/n] - Floor[q/n])^2 + (Mod[p, n] - Mod[q, n])^2;
   (*Triads of points*)
   t0 = Subsets[Range[0, n^2 - 1], {3, 3}];
   (* Exclude collinear vertices *)
   t1 = Select[t0,
     Det[Map[{Floor[#/n], Mod[#, n], 1} &, {#[[1]], #[[2]], #[[
           3]]}]] != 0 &];
   (*Calculate sides*)
   t2 = Map[{#,
       Sort[{{#[[2]], #[[3]]}, {#[[3]], #[[1]]}, {#[[1]], #[[2]]}} /.
         sqDist]} &, t1];
   (*Exclude not-scalenes*)
   t2 = Select[
     t2, #[[2, 1]] != #[[2, 2]] && #[[2, 2]] != #[[2, 3]] && #[[2,
          3]] != #[[2, 1]] &];
   (* Find groups of congruent triangles *)
   t3 = GatherBy[Range[Length[t2]], t2[[#, 2]] &];
   Return[Length[t3]];
   ];
Map[q[#] &, Range[10]] (* César Eliud Lozada, Mar 26 2021 *)

Formula

a(n) = A028419(n) - A189978(n).