A189978 -id:A189978 - cp's OEIS frontend

A028419 Congruence classes of triangles which can be drawn using lattice points in n X n grid as vertices.

0, 1, 8, 29, 79, 172, 333, 587, 963, 1494, 2228, 3195, 4455, 6050, 8032, 10481, 13464, 17014, 21235, 26190, 31980, 38666, 46388, 55144, 65131, 76449, 89132, 103337, 119184, 136757, 156280, 177796, 201430, 227331, 255668, 286606, 320294, 356884, 396376, 439100, 485427, 535049, 588457, 645803

Offset: 0

David J. Rusin

nonn

Ron Knott, Table of n, a(n) for n = 0..60
D. Rusin, Lattice Problem (Triangles) [Dead link]
D. Rusin, Lattice Problem (Triangles) [Cached copy]

Cf. A028492, A189979, A190021, A190022, A189978, A190313, A189978, A190313.

a:=proc(n) local TriangleSet,i,j,k,l,A,B,C; TriangleSet:={}: for i from 0 to n do for j from 0 to n do for k from 0 to n do for l from 0 to n do A:=i^2+j^2: B:=k^2+l^2: C:=(i-k)^2+(j-l)^2: if A^2+B^2+C^2<>2*(A*B+B*C+C*A) then TriangleSet:={op(TriangleSet),sort([sqrt(A),sqrt(B),sqrt(C)])}: fi: od: od: od: od: return(nops(TriangleSet)); end: # Martin Renner, May 03 2011

More terms from Chris Cole (chris(AT)questrel.com), Jun 28 2003

a(36)-a(39) from Martin Renner, May 08 2011

A241237 Number of isosceles triangles, distinct up to congruence, on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 3, 15, 35, 69, 106, 162, 222, 300, 382, 486, 587, 715, 840, 997, 1147, 1313, 1491, 1700, 1890, 2129, 2341, 2598, 2842, 3126, 3394, 3711, 3995, 4341, 4641, 5024, 5349, 5750, 6128, 6540, 6959, 7381, 7772, 8255, 8722, 9252, 9688, 10220, 10698, 11277, 11806, 12381, 12905

Offset: 1

Martin Renner, Apr 17 2014

nonn

A centered hexagonal grid of size n is a grid with A003215(n-1) points forming a hexagonal lattice.

			For n = 2 the three kinds of non-congruent isosceles triangles are the following:
/. *     * *     * .
. * *   . . *   . . *
\. .     . .     * .

Eric Weisstein's World of Mathematics, Hex Number.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A189978, A241228.

a(n) = A241231(n) - A241236(n).

a(7) from Martin Renner, May 31 2014
a(8)-a(21) from Giovanni Resta, May 31 2014
More terms from Bert Dobbelaere, Oct 17 2022

A190313 Number of scalene triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

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

Offset: 1

Martin Renner, May 08 2011

nonn

Eric Weisstein's World of Mathematics, Geoboard.
Eric Weisstein's World of Mathematics, Scalene Triangle.

Cf. A028419, A189978.

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 *)

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

A190309 Number of acute isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 2, 5, 11, 19, 29, 40, 58, 74, 94, 113, 141, 168, 201, 227, 267, 304, 348, 390, 438, 483, 537, 590, 657, 709, 776, 837, 913, 979, 1057, 1130, 1225, 1299, 1396, 1472, 1576, 1663, 1768, 1863, 1974

Offset: 1

Martin Renner, May 08 2011

nonn

Eric Weisstein's World of Mathematics, Geoboard.
Eric Weisstein's World of Mathematics, Acute Triangle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A028419, A108279, A189978, A190021, A190310.

a(n) = A189978(n) - A190310(n) - A108279(n).

A190310 Number of obtuse isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 9, 12, 19, 24, 32, 37, 51, 57, 69, 80, 99, 107, 127, 136, 161, 176, 196, 207, 246, 262, 286, 306, 343, 357, 399, 414, 460, 485, 517, 544, 605, 623, 659, 689, 757

Offset: 1

Martin Renner, May 08 2011

nonn

Eric Weisstein's World of Mathematics, Geoboard.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
Eric Weisstein's World of Mathematics, Isosceles Triangle.

Cf. A028419, A108279, A189978, A190022, A190309.

a(n) = A189978(n) - A190309(n) - A108279(n).

A272053 a(n) is the number of equivalence classes of simple, open polygonal chains consisting of two segments and with all three vertices on the lattice points of an n X n grid.

Original entry on oeis.org

0, 2, 19, 76, 215, 481, 946, 1691, 2789, 4356, 6525, 9397, 13128, 17874, 23768, 31071, 39953, 50551, 63141, 77947, 95234, 115223, 138305, 164501, 194344, 228218, 266165, 308688, 356104, 408731, 467166, 531616, 602362, 679952, 764821, 857517

Offset: 0

Alec Jones, Apr 18 2016

nonn

The chains are counted up to congruence.
Proof that a(n) = 3*A190313(n) + 2*A189978(n):
Let ABC be a lattice triangle in an n X n grid. If ABC is scalene, then the pairs (BA,AC), (AB,BC), and (AC, CB) form three inequivalent polygonal chains; likewise, if ABC is isosceles and AB is the base of the triangle, then (BA,AC) and (AC,CB) form two distinct polygonal chains, while (BC,CA) is congruent to (AB,BC).
Now consider an arbitrary 2-segment polygonal chain (XY,YZ). By the side-angle-side criterion for triangle congruence, the triangle to which XY and YZ belong is determined up to congruence, and so the proposed formula does not over-count. Thus a(n) = 3*A190313(n) + 2*A189978(n).

Alec Jones, Examples for n = 1 to 2

Cf. A190313, A189978.

a(n) = 3*A190313(n) + 2*A189978(n).

cp's OEIS Frontend

A028419 Congruence classes of triangles which can be drawn using lattice points in n X n grid as vertices.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Extensions

A241237 Number of isosceles triangles, distinct up to congruence, on a centered hexagonal grid of size n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A190313 Number of scalene triangles, distinct up to congruence, on an n X n grid (or geoboard).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A190309 Number of acute isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Views

Author

Keywords

Links

Crossrefs

Formula

A190310 Number of obtuse isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Views

Author

Keywords

Links

Crossrefs

Formula

A272053 a(n) is the number of equivalence classes of simple, open polygonal chains consisting of two segments and with all three vertices on the lattice points of an n X n grid.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula