Three sides of right triangles having the hypotenuse and one leg that are triangular numbers.
4, 3, 8, 9, 8, 27, 16, 15, 64, 25, 8, 323, 25, 13, 312, 25, 22, 204, 25, 24, 125, 29, 24, 315, 34, 13, 588, 35, 27, 504, 36, 35, 216, 39, 24, 720, 49, 48, 343, 64, 63, 512, 65, 32, 2079, 81, 80, 729, 100, 68, 4472, 100, 95, 2170, 100, 99, 1000, 105, 63, 5187
1
Let tri(n) be the n-th triangular number n*(n+1)/2. The order of the three numbers is: h (where tri(h) is the hypotenuse), x (where tri(x) is a leg), and y (a leg). The three numbers are shown in different colors in the plot above.
T. D. Noe, Plot of 100 triples
T. D. Noe, Table of 100 triples
Eric W. Weisstein, MathWorld: Triangular Number
(Mma) nn = 1000; t = {}; Do[If[IntegerQ[z = Sqrt[-a^2 (a + 1)^2 + b^2 (b + 1)^2]/2], AppendTo[t, {b, a, z}]], {a, nn}, {b, a+1, nn}]; t = Sort[t]
Cf. S000337, S000338, S000339.
nonn,nice
T. D. Noe, Nov 13 2014