A190312 - OEIS

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A190312

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

0, 0, 40, 368, 1704, 5704, 15400, 36096, 75632, 145968, 263592, 451392, 738360, 1163552, 1774840, 2632344, 3808992, 5394752, 7493936, 10233832, 13759008, 18241312, 23877984, 30896984, 39551456, 50137240, 62983128, 78459880

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Table of n, a(n) for n=1..28.

Eric Weisstein's World of Mathematics, Geoboard.

Eric Weisstein's World of Mathematics, Scalene Triangle.

FORMULA

a(n) = A045996(n) - A186434(n).

MATHEMATICA

q[n_] :=

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

(* 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];

(* Select scalenes *)

t2 = Select[t2,

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

Return[Length[t2]];

];

Map[q[#] &, Range[9]] (* César Eliud Lozada, Mar 26 2021 *)

CROSSREFS

Cf. A045996, A186434.

Sequence in context: A251431 A285919 A234913 * A229532 A252180 A008355

Adjacent sequences: A190309 A190310 A190311 * A190313 A190314 A190315

KEYWORD

nonn

AUTHOR

Martin Renner, May 08 2011

STATUS

approved