Hypotenuses of right triangles having legs that are triangular numbers.

35, 39, 53, 111, 305, 555, 822, 845, 1122, 1189, 1515, 1628, 1639, 1709, 2015, 2103, 2125, 2359, 3653, 3816, 3860, 4375, 6825, 8425, 9181, 10504, 10725, 13530, 16964, 18525, 21822, 26820, 28655, 31285, 37230, 40391, 46189, 48149, 59475, 61974, 73115, 76125

1

This problem came up in MathOverflow. The author wanted all three sides to be triangular. It appears that there is only one such triangle. This sequence was formed by relaxing the requirement on the hypotenuse. Sequence S000338 gives the three sides of the triangle.

T. D. Noe, Plot of 800 terms

T. D. Noe, Table of 800 terms

Ricardo Buring, MathOverflow: How many Pythagorean triples are there in which every member is triangular?

Maciej Ulas, On certain diophantine equations related to triangular and tetrahedral numbers, arXiv 0811.2477 (Nov 15 2008)

Eric W. Weisstein, MathWorld: Triangular Number

Example: The first term, 35, is the hypotenuse of the right triangle whose sides are 21 and 28, the sixth and seventh triangular numbers.

(Mma) nn = 1000; t = {}; mx = 1 + (1/4) nn^2 (nn + 1)^2; Do[z2 = (a^2 (a + 1)^2 + b^2 (b + 1)^2)/4; If[z2 <= mx && IntegerQ[z = Sqrt[z2]], AppendTo[t, {z, a, b}]], {a, nn}, {b, a, nn}]; t = Sort[t]; First /@ t

Cf. S000338, S000339, S000340.

nonn,nice

T. D. Noe, Nov 13 2014, extended Nov 18 2015