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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. 14
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS The triangle must have nonzero area (three collinear points don't count). LINKS 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. 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 EXAMPLE 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. CROSSREFS Rows 2,3,4,5 are A271911, A271912, A271913, A271915. Main diagonal = A186434. Cf. A187452, A271914. Sequence in context: A127774 A127319 A272626 * A249346 A035539 A178517 Adjacent sequences:  A271907 A271908 A271909 * A271911 A271912 A271913 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Apr 24 2016 STATUS approved

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Last modified September 22 02:38 EDT 2021. Contains 347605 sequences. (Running on oeis4.)