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 A190313 Number of scalene triangles, distinct up to congruence, on an n X n grid (or geoboard). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS 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

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Last modified December 6 12:26 EST 2022. Contains 358634 sequences. (Running on oeis4.)