Triangular Squares

2016-07-16, post № 133

programming, Python, Wolfram Language, #equation, #number, #number theory, #numbers, #OEIS, #square, #triangle, #triangles

In a recent video, Matt Parker showed a triangular number that also is a square number, 𝟨, and asked if there were more.

A triangular number has the form \frac{n^2+n}{2} — shown by Euler — and a square number has the form m^2.
Triangular squares are those numbers for which \frac{n^2+n}{2}=m^2 with n,m\in\mathbb{N}.
Examples are \{0,1,6,35,204,1189,6930,\dots\} (sequence A001109 in OEIS).

To check if triangular numbers are square numbers is easy (code listed below), but a mathematical function would be nicer.
The first thing I tried was to define the triangular number’s square root as a whole number, \sqrt{\frac{n^2+n}{2}}=\lfloor\sqrt{\frac{n^2+n}{2}}\rfloor. This function does not return the square numbers that are triangular but the triangular numbers that are square.
The resulting sequence is \{0,1,8,49,288,1681,9800,\dots\} (sequence A001108 in OEIS).

Source code: triangular-squares.py