The numbers that are both square and triangular are beautifully related to the best approximations to √2:
Factors of root
|36||6||2 * 3|
|1225||35||5 * 7|
|41616||204||12 * 17|
and so on.
In each case the factors of the root are the numerator and denominator of the next approximation to √2. — David Wells, The Penguin Dictionary of Curious and Interesting Mathematics (1986), entry for “36”.
• A001110 — Square triangular numbers: numbers that are both triangular and square