Problems & Puzzles: Puzzles

Problem 82.  Pythagorean triangles again -II From the sequences A245646, A245647 and A245648, Ivan Ianakiev sends the following nice puzzle, originally in Bulgarian, alike the Problem 81 The problem 81 was posed by Sierpinski over 70 years ago and no one has found another such triple. I'd like to present a similar problem of mine, which: a) is newer and b) has several visible solutions (see OEIS A245646, A245647, A245648), which might (or might not) suggest they are easier to find:   The Pythagorean triple (3,4,5) consists of the triangular number 3, the square number 4 and the pentagonal number 5. Other such triples are (9,12,15), (100,105,145) and (900,2625,2775).  Q. Can you find more like these four already found? ____ Please notice that the order of the triangular, square and pentagonal integers may vary from triplet to triplet.

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On week 14-20, March 2021, contributions came from Emmanuel Vantieghem,

Emmanuel wrote:

I could not find a fifth solution but I found a Pytaghorean triple

   {482517, 933156, 1050525}

in which  482517  is generalized pentagonal, i.e. : it equals  (3n^2 - n)/2  for  n = -567
933156  is square = 966^2
1050525  is triangular  n(n + 1)/2  with  n = 1449

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Simon Cavegn wrote:

Found no more than the known solutions.
My program's search space was [first 233000 triangular numbers] x [first 1000000 pentagonal numbers]

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