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Number of scalene triangles on an n X n grid (or geoboard).
1

%I #27 Feb 11 2024 20:59:45

%S 0,0,40,368,1704,5704,15400,36096,75632,145968,263592,451392,738360,

%T 1163552,1774840,2632344,3808992,5394752,7493936,10233832,13759008,

%U 18241312,23877984,30896984,39551456,50137240,62983128,78459880

%N Number of scalene triangles 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) = A045996(n) - A186434(n).

%t q[n_] :=

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

%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, 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]]}} /. sqDist]}&, t1];

%t (* Select scalenes *)

%t t2 = Select[t2,

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

%t Return[Length[t2]];

%t ];

%t Map[q[#] &, Range[9]] (* _César Eliud Lozada_, Mar 26 2021 *)

%Y Cf. A045996, A186434.

%K nonn

%O 1,3

%A _Martin Renner_, May 08 2011