A186434 -id:A186434 - cp's OEIS frontend

A271908 a(n) = A186434(n)/4.

0, 1, 9, 37, 111, 266, 550, 1006, 1744, 2821, 4349, 6405, 9203, 12804, 17418, 23164, 30348, 39023, 49591, 62073, 76997, 94454, 114768, 138054, 165176, 196019, 231085, 270557, 315283, 365102, 421116, 482950, 552092, 628401, 712437, 804509, 906217, 1016724, 1137140, 1267836, 1410790, 1564903

Offset: 1

N. J. A. Sloane, Apr 22 2016

nonn

It would be nice to have a formula.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A186434.

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

A271910 Array read by antidiagonals: T(n,k) = number of ways to choose 3 distinct points from an n X k rectangular grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 10, 10, 0, 0, 16, 36, 16, 0, 0, 24, 68, 68, 24, 0, 0, 32, 108, 148, 108, 32, 0, 0, 42, 150, 248, 248, 150, 42, 0, 0, 52, 200, 360, 444, 360, 200, 52, 0, 0, 64, 252, 488, 672, 672, 488, 252, 64, 0, 0, 76, 312, 620, 932, 1064, 932, 620, 312, 76, 0

Offset: 1

N. J. A. Sloane, Apr 24 2016

The triangle must have nonzero area (three collinear points don't count).

			Initial rows of the array:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 4, 10, 16, 24, 32, 42, 52, 64, 76, ...
0, 10, 36, 68, 108, 150, 200, 252, 312, 374, ...
0, 16, 68, 148, 248, 360, 488, 620, 768, 924, ...
0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, ...
0, 32, 150, 360, 672, 1064, 1510, 1984, 2524, 3092, ...
0, 42, 200, 488, 932, 1510, 2200, 2944, 3792, 4690, ...
0, 52, 252, 620, 1204, 1984, 2944, 4024, 5256, 6568, ...
0, 64, 312, 768, 1512, 2524, 3792, 5256, 6976, 8816, ...
0, 76, 374, 924, 1836, 3092, 4690, 6568, 8816, 11284, ...
...
As a triangle:
0,
0, 0,
0, 4, 0,
0, 10, 10, 0,
0, 16, 36, 16, 0,
0, 24, 68, 68, 24, 0,
0, 32, 108, 148, 108, 32, 0,
0, 42, 150, 248, 248, 150, 42, 0,
0, 52, 200, 360, 444, 360, 200, 52, 0,
0, 64, 252, 488, 672, 672, 488, 252, 64, 0,
...
To illustrate T(2,3)=10: Label the points
1 2 3
4 5 6
There are 8 small isosceles triangles like 124 plus 135 and 246, for a total of 10.

Chai Wah Wu, Table of n, a(n) for n = 1..3003
Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.

Rows 2,3,4,5 are A271911, A271912, A271913, A271915.
Main diagonal = A186434.
Cf. A187452, A271914.

It appears that for each n >= 2, there is a number K(n) such that row n satisfies the recurrence a(k) = 2*a(k-1)-2*a(k-3)+a(k-4) for k >= K(n). This is based on the fact that the conjectured generating functions for rows 2, 3, 4, 5 have the same denominator, and on Colin Barker's conjectured recurrence for A271911. K(n) is determined by the degree of the numerator of the g.f.

Above conjecture about the recurrence is true for K(n) = (n-1)^2+4 if n is even and K(n) = (n-1)^2+3 if n is odd and not true for smaller K(n). See paper in links. - Chai Wah Wu, May 07 2016

A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

Original entry on oeis.org

0, 0, 4, 0, 2, 12, 0, 0, 6, 16, 0, 2, 4, 6, 24, 0, 0, 2, 8, 10, 28, 0, 2, 4, 2, 8, 6, 36, 0, 0, 2, 0, 6, 8, 10, 40, 0, 2, 4, 2, 12, 10, 8, 10, 56, 0, 0, 2, 4, 2, 4, 10, 8, 10, 60, 0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60, 0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64, 0, 2

Offset: 1

Lars Blomberg, Feb 16 2017

			Triangle begins:
0
0, 4
0, 2, 12
0, 0, 6, 16
0, 2, 4, 6, 24
0, 0, 2, 8, 10, 28
0, 2, 4, 2, 8, 6, 36
0, 0, 2, 0, 6, 8, 10, 40
0, 2, 4, 2, 12, 10, 8, 10, 56
0, 0, 2, 4, 2, 4, 10, 8, 10, 60
0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60
0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64
0, 2, 4, 2, 4, 6, 8, 10, 16, 14, 12, 14, 72
0, 0, 2, 0, 2, 4, 2, 8, 14, 4, 6, 12, 18, 76
0, 2, 4, 2, 4, 2, 8, 2, 8, 10, 16, 10, 12, 10, 84
0, 0, 2, 0, 6, 4, 2, 4, 6, 16, 6, 4, 10, 12, 14, 88
0, 2, 4, 2, 4, 2, 8, 2, 16, 6, 16, 10, 16, 6, 24, 10, 104
0, 0, 2, 0, 2, 0, 2, 4, 6, 4, 10, 12, 10, 12, 10, 12, 14, 100
0, 2, 4, 2, 4, 2, 12, 6, 4, 6, 12, 10, 20, 6, 12, 14, 16, 10, 124
0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 6, 12, 10, 12, 10, 12, 18, 12, 10, 112
-----
Denote by 'o' the point adjacent to the two equal sides, and by 'x' the other two.
n=4, k=3:
...x  x...  .o..  ..o.  x...  ...x
o...  ...o  ...x  x...  ...x  x...
...x  x...  x...  ...x  .o..  ..o.
So T(4,3)=6.
-----
n=4,k=4:
o...  ...o  .x..  ..x.  o...  ...o  ..x.  .x..
...x  x...  ....  ....  ....  ....  ...x  x...
....  ....  ...x  x...  ...x  x...  ....  ....
.x..  ..x.  o...  ...o  ..x.  .x..  o...  ...o
-
...x  x...  x...  ...x  o..x  x..o  x...  ...x
.o..  ..o.  ....  ....  ....  ....  ....  ....
....  ....  .o..  ..o.  ....  ....  ....  ....
x...  ...x  ...x  x...  x...  ...x  o..x  x..o
So T(4,4)=16.

Lars Blomberg, Table of n, a(n) for n = 1..9870 (the first 140 rows)
Lars Blomberg, Algorithms for computing A279413, A279414, A186434 and A271908

Cf. A186434, A187452, A271910-A271913, A271915, A279414.
See A279415 for right isosceles triangles.
See A280639 for obtuse isosceles triangles.
See A279418 for acute isosceles triangles.
See A279433 for all right triangles.
See A280652 for all obtuse triangles.
See A280653 for all acute triangles.
See A279432 for all triangles.

A189978 Number of isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 5, 11, 22, 35, 53, 70, 100, 126, 159, 188, 237, 276, 328, 372, 439, 491, 564, 623, 706, 775, 859, 931, 1049, 1129, 1231, 1323, 1448, 1540, 1674, 1772, 1928, 2041, 2183, 2301, 2483, 2602, 2758, 2898, 3095

Offset: 1

Martin Renner, May 03 2011

			For n=3 the five isosceles triangles are:
**.  *.*  .*.  ..*  *..
*..  ...  *..  *..  ..*
...  *..  .*.  ..*  .*.

Alec Jones, Examples for n = 1 to 5
Alec Jones, Example for n = 24

Cf. A028419, A186434.

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

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

A241228 Number of isosceles triangles on a centered hexagonal grid of size n.

Original entry on oeis.org

0, 20, 252, 1296, 4232, 10668, 22956, 43832, 77160, 126972, 198176, 296976, 429252, 602876, 825552, 1106376, 1454192, 1879956, 2397024, 3014312, 3747564, 4609476

Offset: 1

Martin Renner, Apr 17 2014

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, Isosceles Triangle.

Cf. A186434.

a(n) = A241223(n) - A241227(n).

a(7) from Martin Renner, May 31 2014

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

A189416 Number of parallelograms on an n X n grid.

Original entry on oeis.org

0, 1, 22, 158, 674, 2159, 5664, 13004, 26904, 51401, 92094, 156710, 255090, 400359, 608656, 900100, 1299336, 1836461, 2546550, 3472162, 4661898, 6173123, 8071952, 10434600, 13346080, 16905033, 21221558, 26419338, 32636098, 40027283, 48761448

Offset: 1

Martin Renner, Apr 21 2011

nonn

Nathaniel Johnston, C program for computing terms
Eric Weisstein's World of Mathematics, Parallelogram

Cf. A186434, A189413, A189414.

a[n_] := Sum[(n-a)*(n-b)*(2*a*b - GCD[a, b]), {a, 1, n-1}, {b, 1, n-1}];
Array[a, 31] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

a(n) = sum(a=1, n-1, sum(b=1, n-1, (n-a)*(n-b)*(2*a*b - gcd(a,b)) )); \\ Andrew Howroyd, Sep 19 2017

a(n) = Sum_{a=1..n-1} Sum_{b=1..n-1} (n-a)*(n-b)*(2*a*b - gcd(a,b)). - Andrew Howroyd, Sep 19 2017

a(6)-a(31) from Nathaniel Johnston, Apr 24 2011

A190317 Number of acute isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 8, 48, 164, 448, 976, 1864, 3328, 5512, 8640, 12984, 18836, 26576, 36584, 49056, 64560, 83640, 106904, 134816, 168004, 206952, 252480, 305360, 366312, 436120, 515864, 605928, 707644, 822120, 950216, 1092784, 1251784, 1427848, 1622416, 1835968, 2070404

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A279418 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) * A279418(i,j))) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 100 terms from Chai Wah Wu)

Cf. A186434, A187452, A190318, A279418.

a(n) = A186434(n) - A187452(n) - A190318(n).

a(10)-a(37) from Nathaniel Johnston, May 09 2011

A190318 Number of obtuse isosceles triangles on an n X n grid.

Original entry on oeis.org

0, 0, 0, 4, 36, 100, 256, 496, 968, 1672, 2736, 4092, 6188, 8764, 12144, 16464, 22224, 28928, 37400, 47076, 59244, 73580, 90344, 109000, 132048, 158000, 187528, 220716, 259348, 301388, 350088, 402792, 463176, 529720, 602888, 683092, 774476, 872100, 978232

Offset: 1

Martin Renner, May 08 2011

nonn

Place all bounding boxes of A280639 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) * A280639(i,j))) where k=1 when i=j and k=2 otherwise. - Lars Blomberg, Mar 02 2017

Lars Blomberg, Table of n, a(n) for n = 1..10000 (the first 100 terms from Chai Wah Wu)
Nathaniel Johnston, C program for computing terms

Cf. A186434, A187452, A190317, A280639.

a(n) = A186434(n) - A190317(n) - A187452(n).

a(10)-a(39) from Nathaniel Johnston, May 09 2011

A271913 Number of ways to choose three distinct points from a 4 X n grid so that they form an isosceles triangle.

Original entry on oeis.org

0, 16, 68, 148, 248, 360, 488, 620, 768, 924, 1096, 1272, 1464, 1660, 1872, 2088, 2320, 2556, 2808, 3064, 3336, 3612, 3904, 4200, 4512, 4828, 5160, 5496, 5848, 6204, 6576, 6952, 7344, 7740, 8152, 8568, 9000, 9436, 9888, 10344, 10816, 11292, 11784, 12280, 12792, 13308, 13840, 14376, 14928, 15484

Offset: 1

N. J. A. Sloane, Apr 24 2016

nonn

Chai Wah Wu, Counting the number of isosceles triangles in rectangular regular grids, arXiv:1605.00180 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -2, 1).

Row 4 of A271910.

Cf. A186434, A187452.

Join[{0, 16, 68, 148, 248, 360, 488, 620}, LinearRecurrence[{2, 0, -2, 1}, {768, 924, 1096, 1272}, 42]] (* Jean-François Alcover, Sep 03 2018 *)

Conjectured g.f.: 4*x*(x^10-x^8+2*x^6+x^5+4*x^4+4*x^3-3*x^2-9*x-4)/((x+1)*(x-1)^3).
Conjectured recurrence: a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n > 12.
Conjectures from Colin Barker, Apr 25 2016: (Start)
a(n) = -3/2*(143+(-1)^n)+64*n+5*n^2 for n>8.
a(n) = 5*n^2+64*n-216 for n>8 and even.
a(n) = 5*n^2+64*n-213 for n>8 and odd.
(End)
The conjectured g.f. and recurrence are true. See paper in links. - Chai Wah Wu, May 07 2016

More terms from Jean-François Alcover, Sep 03 2018

cp's OEIS Frontend

A271908 a(n) = A186434(n)/4.

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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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A271910 Array read by antidiagonals: T(n,k) = number of ways to choose 3 distinct points from an n X k rectangular grid so that they form an isosceles triangle.

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A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

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A189978 Number of isosceles triangles, distinct up to congruence, on an n X n grid (or geoboard).

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Maple

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A241228 Number of isosceles triangles on a centered hexagonal grid of size n.

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A189416 Number of parallelograms on an n X n grid.

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A190317 Number of acute isosceles triangles on an n X n grid.

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A190318 Number of obtuse isosceles triangles on an n X n grid.

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