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

1,3

Links

Table of n, a(n) for n=1..41.

Eric Weisstein's World of Mathematics, Geoboard.

Eric Weisstein's World of Mathematics, Scalene Triangle.

Formula

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

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 *)

Crossrefs

Cf. A028419, A189978.

Sequence in context: A222204 A027289 A061317 * A139362 A012763 A006011

Adjacent sequences: A190310 A190311 A190312 * A190314 A190315 A190316

Keyword

nonn

Author

Martin Renner, May 08 2011

Status

approved