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A190313
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Number of scalene triangles, distinct up to congruence, on an n X n grid (or geoboard).
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3
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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
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..41.
Eric Weisstein's World of Mathematics, Geoboard.
Eric Weisstein's World of Mathematics, Scalene Triangle.
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FORMULA
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a(n) = A028419(n) - A189978(n).
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A028419, A189978.
Sequence in context: A222204 A027289 A061317 * A139362 A012763 A006011
Adjacent sequences: A190310 A190311 A190312 * A190314 A190315 A190316
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KEYWORD
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nonn
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AUTHOR
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Martin Renner, May 08 2011
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STATUS
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approved
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