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 1-3 of 3 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

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

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, where the angle at B is a right angle. The triangles can have any orientation.

Examples

			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

Links

Crossrefs

Cf. A045996, A077435, A186434, A189416.

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:
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:
  • Mathematica
    LinearRecurrence[{4,-5,0,5,-4,1},{0,4,28,96,244,516},40] (* Harvey P. Dale, Apr 29 2016 *)
  • PARI
    concat(0, Vec(4*x^2*(1+3*x+x^2)/((1-x)^5*(1+x)) + O(x^50))) \\ Colin Barker, Apr 25 2016

Formula

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)

Extensions

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

A189415 Number of trapezoids on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 50, 490, 2618, 9519, 28432, 70796, 157912, 321161, 610482, 1082570, 1848362, 3003015, 4716792, 7204604, 10730528, 15530189, 22093410, 30723078, 42146178, 56981411, 75952240, 99685104, 129757248
Offset: 1

Views

Author

Martin Renner, Apr 21 2011

Keywords

Links

Crossrefs

Cf. A181945, A186434, A189413, A189414, A189416, A189418.

Extensions

a(6)-a(25) from Nathaniel Johnston, Apr 25 2011
Showing 1-3 of 3 results.

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