cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A186434 Number of 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, 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

Views

Author

Martin Renner, Apr 10 2011, Apr 13 2011

Keywords

Comments

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

Links

Crossrefs

Cf. A045996, A077435, A187452, A189416, A271910-A271913, A271915, A279413, A279414.
Dividing by 4 gives A271908.

Programs

  • Maple
    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:

Extensions

a(10)-a(33) 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

Views

Author

N. J. A. Sloane, Apr 24 2016

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Formula

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

Views

Author

Lars Blomberg, Feb 16 2017

Keywords

Examples

			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.

Links

Crossrefs

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.

A279414 a(n) is the total number of isosceles triangles having a bounding box n X k where k is in the range 1 to n inclusive.

Original entry on oeis.org

0, 4, 14, 22, 36, 48, 58, 66, 104, 100, 110, 118, 164, 148, 174, 174, 232, 200, 266, 226, 300, 272, 290, 282, 412, 332, 362, 358, 440, 376, 494, 386, 572, 464, 490, 490, 660, 476, 546, 562, 756, 552, 718, 582, 760, 692, 682, 634, 1004, 716, 862, 746, 900, 744
Offset: 1

Views

Author

Lars Blomberg, Feb 16 2017

Keywords

Comments

Row sums of A279413.

Links

Crossrefs

Cf. A186434, A187452, A271910-A271913, A271915, A279413.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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