A190021 -id:A190021 - 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

A190022 Number of obtuse triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 2, 12, 39, 95, 193, 355, 597, 943, 1426, 2071, 2904, 3977, 5306, 6956, 8963, 11370, 14225, 17587, 21515, 26053, 31310, 37282, 44061, 51785, 60436, 70127, 80939, 92952, 106267, 120982, 137124, 154841, 174225, 195366, 218394, 243457, 270505, 299749, 331441

Offset: 1

Martin Renner, May 04 2011

nonn

			For n = 3 the two obtuse triangles are:
*..   *..
*..   *..
.*.   ..*

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

Cf. A028419, A189979, A190021.

Cf. A103428.

Triangles:=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(TriangleSet); end:
IsObtuseTriangle:=proc(T) if T[1]^2+T[2]^2

a(n) = A028419(n) - A190021(n) - A189979(n).

a(21)-a(40) from Martin Renner, May 08 2011

A241232 Number of acute triangles, distinct up to congruence, on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 2, 14, 49, 134, 296, 580, 1034, 1720, 2691, 4043, 5841, 8193, 11178, 14935, 19567, 25197, 31954, 40006, 49521, 60596, 73442, 88238, 105158, 124432, 146220, 170802, 198278, 228999, 263185, 300988, 342775, 388775, 439269, 494462, 554839, 620474, 691717, 769060, 852639

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 two kinds of non-congruent acute triangles are the following:
/. *     * .
. * *   . . *
\. .     * .

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

Cf. A190021, A241224.

a(n) = A241231(n) - A241233(n) - A241234(n)

a(7) from Martin Renner, May 31 2014
a(8)-a(18) from Giovanni Resta, May 31 2014
More terms from Bert Dobbelaere, Oct 17 2022

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).

cp's OEIS Frontend

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

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A190022 Number of obtuse triangles, distinct up to congruence, on an n X n grid (or geoboard).

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A241232 Number of acute triangles, distinct up to congruence, on a centered hexagonal grid of size n.

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

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