## Triangular Squares

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).

# Python 2.7.7 Code
# Jonathan Frech 13th of July, 2016
#         edited 15th of July, 2016

## Rotating Squares

Using the pygame.transform.rotozoom() function, a velocity argument and changing colors, this program rotates squares across the screen.

#### Controls

• Left click spawns in a new square at current mouse position
• Space takes a screenshot

# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 27th of November, 2015
#         edited  2nd of January , 2016