A077435 -id:A077435 - cp's OEIS frontend

A186434 Number of isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

0, 4, 36, 148, 444, 1064, 2200, 4024, 6976, 11284, 17396, 25620, 36812, 51216, 69672, 92656, 121392, 156092, 198364, 248292, 307988, 377816, 459072, 552216, 660704, 784076, 924340, 1082228, 1261132, 1460408, 1684464, 1931800, 2208368

Offset: 1

Martin Renner, Apr 10 2011, Apr 13 2011

nonn

This counts triples of distinct points A,B,C such that A,B,C are the vertices of an isosceles triangle with nonzero area. It would be nice to have a formula. - N. J. A. Sloane, Apr 22 2016

Place all bounding boxes of A279413 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A279413(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Feb 20 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 67 terms from Nathaniel Johnston)
Margherita Barile, MathWorld -- Geoboard.
Nathaniel Johnston, C program for computing terms

Cf. A045996, A077435, A187452, A189416, A271910-A271913, A271915, A279413, A279414.

Dividing by 4 gives A271908.

with(linalg):
IsTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if evalf(norm(a,2)+norm(b,2))>evalf(norm(c,2)) and evalf(norm(a,2)+norm(c,2))>evalf(norm(b,2)) and evalf(norm(b,2)+norm(c,2))>evalf(norm(a,2)) then true: else false: fi: end:
IsIsoscelesTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if IsTriangle(points) then if norm(a,2)=norm(b,2) or norm(a,2)=norm(c,2) or norm(b,2)=norm(c,2) then true: else false: fi: else false: fi; end:
a:=proc(n) local P,TriangleSet,i,j,a,b,c; P:=[]: for i from 0 to n do for j from 0 to n do P:=[op(P),[i,j]]: od; od; TriangleSet:={}: for a from 1 to nops(P) do for b from a+1 to nops(P) do for c from b+1 to nops(P) do if IsIsoscelesTriangle([P[a],P[b],P[c]]) then TriangleSet:={op(TriangleSet),[P[a],P[b],P[c]]}; fi; od; od; od; return(nops(TriangleSet)): end:

a(10)-a(33) from Nathaniel Johnston, Apr 25 2011

A187452 Number of right isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

Original entry on oeis.org

0, 4, 28, 96, 244, 516, 968, 1664, 2680, 4100, 6020, 8544, 11788, 15876, 20944, 27136, 34608, 43524, 54060, 66400, 80740, 97284, 116248, 137856, 162344, 189956, 220948, 255584, 294140, 336900, 384160, 436224, 493408, 556036, 624444, 698976, 779988, 867844

Offset: 1

Martin Renner, Apr 10 2011, Apr 13 2011

This counts triples of distinct points A,B,C such that A,B,C are the vertices of an isosceles triangle with nonzero area, where the angle at B is a right angle. The triangles can have any orientation.

			For n=2 if the four points are labeled
ab
cd
then the triangles are abc, abd, acd, bcd,
so a(2)=4.
For n=3, label the points
abc
def
ghi
The triangles are: abd (4*4 ways), acg (4 ways), ace and dbf (4 ways each), for a total of a(3) = 28. - _N. J. A. Sloane_, Jun 30 2016

Nathaniel Johnston and Colin Barker, Table of n, a(n) for n = 1..1000 [first 73 terms from Nathaniel Johnston]
Margherita Barile, MathWorld -- Geoboard.
Jessica Gonzalez, Illustration of a(3)=28
Nathaniel Johnston, C program for computing terms
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A045996, A077435, A186434, A189416.

with(linalg):
IsTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if evalf(norm(a,2)+norm(b,2))>evalf(norm(c,2)) and evalf(norm(a,2)+norm(c,2))>evalf(norm(b,2)) and evalf(norm(b,2)+norm(c,2))>evalf(norm(a,2)) then true: else false: fi: end:
IsRectangularTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if IsTriangle(points) then if dotprod(a,b)=0 or dotprod(a,c)=0 or dotprod(b,c)=0 then true: else false: fi: else false: fi; end:
IsIsoscelesTriangle:=proc(points) local a,b,c; a:=points[3]-points[2]: b:=points[3]-points[1]: c:=points[2]-points[1]: if IsTriangle(points) then if norm(a,2)=norm(b,2) or norm(a,2)=norm(c,2) or norm(b,2)=norm(c,2) then true: else false: fi: else false: fi; end:
IsRectangularIsoscelesTriangle:=proc(points) if IsRectangularTriangle(points) and IsIsoscelesTriangle(points) then true: else false: fi: end:
a:=proc(n) local P,TriangleSet,i,j,a,b,c; P:=[]: for i from 0 to n do for j from 0 to n do P:=[op(P),[i,j]]: od; od; TriangleSet:={}: for a from 1 to nops(P) do for b from a+1 to nops(P) do for c from b+1 to nops(P) do if IsRectangularIsoscelesTriangle([P[a],P[b],P[c]]) then TriangleSet:={op(TriangleSet),[P[a],P[b],P[c]]}; fi; od; od; od; return(nops(TriangleSet)): end:

LinearRecurrence[{4,-5,0,5,-4,1},{0,4,28,96,244,516},40] (* Harvey P. Dale, Apr 29 2016 *)

concat(0, Vec(4*x^2*(1+3*x+x^2)/((1-x)^5*(1+x)) + O(x^50))) \\ Colin Barker, Apr 25 2016

Empirical: a(n)=4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6). [R. H. Hardin, Apr 30 2011]
Empirical g.f.: 4*x*(x^2+3*x+1)/((1+x)*(1-x)^5). - N. J. A. Sloane, Apr 12 2016
Both the recurrence and the g.f. are true. For proof see [Paper in preparation]. - Warren D. Smith, Apr 17 2016
From Colin Barker, Apr 25 2016: (Start)
a(n) = (3-3*(-1)^n-16*n^2+10*n^4)/24.
a(n) = (5*n^4-8*n^2)/12 for n even.
a(n) = (5*n^4-8*n^2+3)/12 for n odd.
(End)

a(10) - a(36) from Nathaniel Johnston, Apr 25 2011

A279433 Triangle read by rows: T(n,k), n>=k>=1, is the number of right triangles with integral coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 6, 4, 0, 4, 12, 4, 0, 4, 6, 12, 12, 0, 4, 8, 12, 12, 4, 0, 4, 4, 6, 12, 20, 4, 0, 4, 4, 12, 12, 12, 20, 4, 0, 4, 4, 4, 14, 12, 20, 12, 12, 0, 4, 4, 4, 12, 12, 16, 12, 12, 20, 0, 4, 4, 8, 8, 6, 12, 20, 20, 20, 4, 0, 4, 4, 4, 4, 12, 28, 12, 12, 12

Offset: 1

Lars Blomberg, Feb 27 2017

			Triangle begins:
0
0,4
0,6,4
0,4,12,4
0,4,6,12,12
0,4,8,12,12,4
0,4,4,6,12,20,4
0,4,4,12,12,12,20,4
0,4,4,4,14,12,20,12,12
0,4,4,4,12,12,16,12,12,20
0,4,4,8,8,6,12,20,20,20,4
0,4,4,4,4,12,28,12,12,12,20,4
0,4,4,4,4,12,6,20,20,16,20,20,12
0,4,4,4,12,4,24,12,12,12,20,12,20,4
0,4,4,4,4,4,12,6,28,20,12,20,20,20,4
0,4,4,4,4,4,8,12,20,20,12,20,12,20,28,4
0,4,4,4,4,12,4,12,18,12,20,12,28,12,20,20,28
-----
The right angle is 'o'.
For n=2, k=2:
ox   xo   x.   .x
x.   .x   ox   xo
So T(2,2)=4
-----
For n=3, k=2:
o.x   x.x   x.o   x..   .o.   ..x
x..   .o.   ..x   o.x   x.x   x.o
So T(3,2)=6

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)

Cf. A077435.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279413 for all isosceles triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A189814 T(n,k)=Number of right triangles on a (n+1)X(k+1) grid.

Original entry on oeis.org

4, 14, 14, 28, 44, 28, 46, 94, 94, 46, 68, 158, 200, 158, 68, 94, 238, 342, 342, 238, 94, 124, 330, 524, 596, 524, 330, 124, 158, 434, 732, 926, 926, 732, 434, 158, 196, 550, 972, 1308, 1444, 1308, 972, 550, 196, 238, 678, 1236, 1754, 2060, 2060, 1754, 1236, 678

Offset: 1

R. H. Hardin Apr 28 2011

Table starts
...4..14...28...46...68...94...124...158...196...238...284...334...388...446
..14..44...94..158..238..330...434...550...678...818...970..1134..1310..1498
..28..94..200..342..524..732...972..1236..1524..1840..2180..2544..2932..3344
..46.158..342..596..926.1308..1754..2250..2794..3390..4026..4702..5426..6190
..68.238..524..926.1444.2060..2784..3596..4492..5470..6516..7630..8820.10070
..94.330..732.1308.2060.2960..4032..5250..6604..8082..9684.11388.13220.15144
.124.434..972.1754.2784.4032..5520..7224..9128.11218.13500.15938.18568.21328
.158.550.1236.2250.3596.5250..7224..9496.12044.14860.17948.21266.24852.28634
.196.678.1524.2794.4492.6604..9128.12044.15332.18990.23012.27354.32052.37032
.238.818.1840.3390.5470.8082.11218.14860.18990.23596.28678.34190.40166.46522

			Some solutions for n=3 k=3
..2..3....2..1....0..2....0..1....3..1....1..3....4..2....2..2....1..1....3..2
..1..2....0..3....0..3....0..2....1..3....1..2....2..1....2..1....0..2....1..3
..4..1....3..2....4..2....5..1....5..3....4..3....5..0....5..2....2..2....2..0

R. H. Hardin, Table of n, a(n) for n = 1..10025

Column 1 is -A147973(n+4)

Diagonal is A077435(n+1)

Empirical for columns
k=1: a(n) = 2*n^2 + 4*n - 2
k=2: a(n) = 6*n^2 + 26*n - 42 for n>3
k=3: a(n) = 12*n^2 + 88*n - 240 for n>8
k=4: a(n) = 20*n^2 + 228*n - 930 for n>15
k=5: a(n) = 30*n^2 + 468*n - 2478 for n>24
k=6: a(n) = 42*n^2 + 886*n - 6080 for n>35
k=7: a(n) = 56*n^2 + 1480*n - 12216 for n>48
k=8: a(n) = 72*n^2 + 2344*n - 23112 for n>63
k=9: a(n) = 90*n^2 + 3516*n - 40434 for n>80
k=10: a(n) = 110*n^2 + 5090*n - 67626 for n>99
k=11: a(n) = 132*n^2 + 7016*n - 105016 for n>120
k=12: a(n) = 156*n^2 + 9564*n - 162094 for n>143
k=13: a(n) = 182*n^2 + 12572*n - 236518 for n>168
k=14: a(n) = 210*n^2 + 16230*n - 337676 for n>195

A190019 Number of acute triangles on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 8, 80, 404, 1392, 3880, 9208, 19536, 38096, 69288, 119224, 196036, 310008, 474336, 705328, 1023216, 1451904, 2020232, 2762848, 3719420, 4937200, 6469424, 8378184, 10734664, 13618168, 17119288, 21338760, 26390452, 32400592, 39508656, 47870200, 57655752

Offset: 1

Martin Renner, May 04 2011

nonn

Place all bounding boxes of A280653 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A280653(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Feb 26 2017

According to Langford (p. 243), the leading order is (53/150-Pi/40)*C(n^2,3). See A093072. - Michael R Peake, Jan 15 2021

Lars Blomberg, Table of n, a(n) for n = 1..5000
Margherita Barile, Geoboard.
Eric Langford, A problem in geometric probability, Mathematics Magazine, Nov-Dec, 1970, 237-244.
Eric Weisstein's World of Mathematics, Acute Triangle.

Cf. A045996, A077435, A093072, A280653.

Cf. A103429 (analogous problem on a 3-dimensional grid).

a(n) = A045996(n) - A077435(n) - A190020(n).

Extended by Ray Chandler, May 04 2011

More terms from Lars Blomberg, Feb 26 2017

A190020 Number of obtuse triangles on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 24, 236, 1148, 3932, 10760, 25392, 53576, 103824, 188104, 322852, 529116, 835028, 1275360, 1893496, 2742208, 3886568, 5402448, 7381316, 9928860, 13168484, 17243896, 22319864, 28579720, 36237928, 45532720, 56732668

Offset: 1

Martin Renner, May 04 2011

nonn

Place all bounding boxes of A280652 that will fit into the n X n grid in all possible positions, and the proper rectangles in two orientations: a(n) = Sum_{i=1..n} Sum_{j=1..i} k * (n-i+1) * (n-j+1) * A280652(i,j) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

According to Langford (p. 243), the leading order is (97/150 + Pi/40)*C(n^2,3). See A093072. - Michael R Peake, Jan 15 2021

Lars Blomberg, Table of n, a(n) for n = 1..10000
Margherita Barile, MathWorld -- Geoboard.
Eric Langford, A problem in geometric probability, Mathematics Magazine, Nov-Dec, 1970, 237-244.
Eric Weisstein's World of Mathematics, Obtuse Triangle.

Cf. A045996, A077435, A093072, A190019, A280652.

a(n) = A045996(n) - A077435(n) - A190019(n).

Extended by Ray Chandler, May 04 2011

A189979 Number of right triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 4, 9, 17, 26, 39, 53, 71, 91, 114, 136, 169, 197, 231, 267, 310, 346, 397, 437, 492, 548, 606, 654, 729, 791, 858, 928, 1007, 1071, 1173, 1241, 1333, 1423, 1509, 1600, 1728, 1814, 1912, 2015, 2144

Offset: 1

Martin Renner, May 03 2011

			For n=3 the four right triangles are:
**.  *.*  *.*  .*.
*..  *..  ...  *..
...  ...  *..  .*.

Cf. A028419, A077435.

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:
IsRectangularTriangle:=proc(T) if T[1]^2+T[2]^2=T[3]^2 or T[1]^2+T[3]^2=T[2]^2 or T[2]^2+T[3]^2=T[1]^2 then true else false fi: end:
a:=proc(n) local TriangleSet,RectangularTriangleSet,i; TriangleSet:=Triangles(n): RectangularTriangleSet:={}: for i from 1 to nops(TriangleSet) do if IsRectangularTriangle(TriangleSet[i]) then RectangularTriangleSet:={op(RectangularTriangleSet),TriangleSet[i]} fi: od: return(nops(RectangularTriangleSet)); end:

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

A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

Original entry on oeis.org

0, 0, 4, 24, 84, 224, 516, 1068, 2016, 3528, 5832, 9256, 14208, 21180, 30728, 43488, 60192, 81660, 108828, 142764, 184708, 236088, 298476, 373652, 463524, 570228, 696012, 843312, 1014720, 1213096, 1441512, 1703352, 2002196, 2341848, 2726400, 3160272, 3648180

Offset: 0

Peter Kagey, May 06 2020

nonn

a(n) >= 4 * A269747(n).
a(n) >= 4 * A000389(n+3) = A210569(n+2).
a(n) >= 4 * (n-1) + 4 * a(n-1) - 6 * a(n-2) + 4 * a(n-3) - a(n-4) for n >= 4.

Peter Kagey, Table of n, a(n) for n = 0..1000 (Computed via Anders Kaseorg's program in the Stack Exchange link.)
Code Golf Stack Exchange, Triangles in a tetrahedron

Cf. A000292, A000389, A210569.
Cf. A000332 (equilateral triangles in triangular grid), A269747 (regular tetrahedra in a tetrahedral grid), A102698 (equilateral triangles in cube), A103158 (regular tetrahedra in cube).
Cf. A077435, A085582, A187452, A189412-A189418, A271910.

A241225 Number of right triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 12, 216, 1104, 3708, 9396, 20304, 38868, 68364, 112632, 176076, 263832, 381924, 536424, 735240, 985896, 1296540, 1676508, 2137392, 2689248, 3344244, 4114020

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.

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

Cf. A077435.

a(n) = A241223(n) - A241224(n) - A241226(n).

a(7) from Martin Renner, May 31 2014

a(8)-a(22) from Giovanni Resta, May 31 2014

A334881 Number of squares in 3-dimensional space whose four vertices have coordinates (x,y,z) in the set {1,...,n}.

Original entry on oeis.org

0, 0, 6, 54, 240, 810, 2274, 5304, 10752, 19992, 34854, 57774, 91200, 139338, 206394, 296832, 417120, 575556, 779238, 1037514, 1359792, 1760694, 2251362, 2845140, 3554976, 4404876, 5416278, 6605946, 7996896, 9621678, 11500962, 13667772, 16143552, 18973608, 22190406

Offset: 0

Peter Kagey, May 14 2020

nonn

a(n) >= 3*n*A002415(n).

			For n = 5, one such square has vertex set {(2,1,1), (5,4,1), (4,5,5), (1,2,5)}.

Zachary Kaplan, Table of n, a(n) for n = 0..100
Zachary Kaplan, Python program
Mathematics Stack Exchange user Olivier Massicot, How many squares in a three-dimensional n X n X n cartesian grid?

Cf. A002415 (squares in square grid), A098928 (cubes in cube grid).

Cf. A000332, A077435, A085582, A102698, A103158, A187452, A189412-A189418, A269747, A271910, A334581.

a(7)-a(12) from Pontus von Brömssen, May 15 2020
a(13)-a(20) from Peter Kagey, Jul 29 2020 via Mathematics Stack Exchange link
Terms a(21) and beyond from Zachary Kaplan, Sep 01 2020, via Mathematics Stack Exchange link

cp's OEIS Frontend

A186434 Number of isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

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A187452 Number of right isosceles triangles that can be formed from the n^2 points of n X n grid of points (or geoboard).

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A279433 Triangle read by rows: T(n,k), n>=k>=1, is the number of right triangles with integral coordinates that have a bounding box of size n X k.

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A189814 T(n,k)=Number of right triangles on a (n+1)X(k+1) grid.

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A190019 Number of acute triangles on an n X n grid (or geoboard).

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A190020 Number of obtuse triangles on an n X n grid (or geoboard).

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A189979 Number of right triangles, distinct up to congruence, on an n X n grid (or geoboard).

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A334581 Number of ways to choose 3 points that form an equilateral triangle from the A000292(n) points in a regular tetrahedral grid of side length n.

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A241225 Number of right triangles on a centered hexagonal grid of size n.

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