A190313 - OEIS

%I #23 May 28 2021 20:54:11

%S 0,0,3,18,57,137,280,517,863,1368,2069,3007,4218,5774,7704,10109,

%T 13025,16523,20671,25567,31274,37891,45529,54213,64082,75320,87901,

%U 102014,117736,135217,154606,176024,199502,225290,253485,284305,317811,354282,393618,436202,482332

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Geoboard.html">Geoboard</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ScaleneTriangle.html">Scalene Triangle</a>.

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

%t q[n_] :=

%t   Module[{sqDist, t0, t1, t2, t3},

%t    (*Squared distances*)

%t    sqDist = {p_,q_} :> (Floor[p/n] - Floor[q/n])^2 + (Mod[p, n] - Mod[q, n])^2;

%t    (*Triads of points*)

%t    t0 = Subsets[Range[0, n^2 - 1], {3, 3}];

%t    (* Exclude collinear vertices *)

%t    t1 = Select[t0,

%t      Det[Map[{Floor[#/n], Mod[#, n], 1} &, {#[[1]], #[[2]], #[[

%t            3]]}]] != 0 &];

%t    (*Calculate sides*)

%t    t2 = Map[{#,

%t        Sort[{{#[[2]], #[[3]]}, {#[[3]], #[[1]]}, {#[[1]], #[[2]]}} /.

%t          sqDist]} &, t1];

%t    (*Exclude not-scalenes*)

%t    t2 = Select[

%t      t2, #[[2, 1]] != #[[2, 2]] && #[[2, 2]] != #[[2, 3]] && #[[2,

%t           3]] != #[[2, 1]] &];

%t    (* Find groups of congruent triangles *)

%t    t3 = GatherBy[Range[Length[t2]], t2[[#, 2]] &];

%t    Return[Length[t3]];

%t    ];

%t Map[q[#] &, Range[10]] (* _César Eliud Lozada_, Mar 26 2021 *)

%Y Cf. A028419, A189978.

%K nonn

%O 1,3

%A _Martin Renner_, May 08 2011