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

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

Author

Martin Renner, Apr 10 2011, Apr 13 2011

Keywords

• nonn
• easy

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

• 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).

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