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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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%I #36 May 09 2016 23:45:28

%S 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,

%T 32,0,0,42,150,248,248,150,42,0,0,52,200,360,444,360,200,52,0,0,64,

%U 252,488,672,672,488,252,64,0,0,76,312,620,932,1064,932,620,312,76,0

%N 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.

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

%H Chai Wah Wu, <a href="/A271910/b271910.txt">Table of n, a(n) for n = 1..3003</a>

%H Chai Wah Wu, <a href="http://arxiv.org/abs/1605.00180">Counting the number of isosceles triangles in rectangular regular grids</a>, arXiv:1605.00180 [math.CO], 2016.

%F 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.

%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

%e Initial rows of the array:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 4, 10, 16, 24, 32, 42, 52, 64, 76, ...

%e 0, 10, 36, 68, 108, 150, 200, 252, 312, 374, ...

%e 0, 16, 68, 148, 248, 360, 488, 620, 768, 924, ...

%e 0, 24, 108, 248, 444, 672, 932, 1204, 1512, 1836, ...

%e 0, 32, 150, 360, 672, 1064, 1510, 1984, 2524, 3092, ...

%e 0, 42, 200, 488, 932, 1510, 2200, 2944, 3792, 4690, ...

%e 0, 52, 252, 620, 1204, 1984, 2944, 4024, 5256, 6568, ...

%e 0, 64, 312, 768, 1512, 2524, 3792, 5256, 6976, 8816, ...

%e 0, 76, 374, 924, 1836, 3092, 4690, 6568, 8816, 11284, ...

%e ...

%e As a triangle:

%e 0,

%e 0, 0,

%e 0, 4, 0,

%e 0, 10, 10, 0,

%e 0, 16, 36, 16, 0,

%e 0, 24, 68, 68, 24, 0,

%e 0, 32, 108, 148, 108, 32, 0,

%e 0, 42, 150, 248, 248, 150, 42, 0,

%e 0, 52, 200, 360, 444, 360, 200, 52, 0,

%e 0, 64, 252, 488, 672, 672, 488, 252, 64, 0,

%e ...

%e To illustrate T(2,3)=10: Label the points

%e 1 2 3

%e 4 5 6

%e There are 8 small isosceles triangles like 124 plus 135 and 246, for a total of 10.

%Y Rows 2,3,4,5 are A271911, A271912, A271913, A271915.

%Y Main diagonal = A186434.

%Y Cf. A187452, A271914.

%K nonn,tabl

%O 1,5

%A _N. J. A. Sloane_, Apr 24 2016