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 — shown by Euler — and a square number has the form .

Triangular squares are those numbers for which with .

Examples are (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, . This function does not return the square numbers that are triangular but the triangular numbers that are square.

The resulting sequence is (sequence A001108 in OEIS).

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

```
# Triangular square numbers
# a*(1+a)/2 == b**2
# Mathematica Code
# n=0;p=1;While[True,n+=p;p++;If[Mod[Sqrt[n],1]==0,Print[n]]]
# OEIS
# http://oeis.org/A001110
# def a(n):
# if n in [1, 0]:
# return n
# return 6*a(n-1)-a(n-2)
# Mathematica Code
# a[0]=0;a[1]=1;a[n_]:=6*a[n-1]-a[n-2]
# import math module
import math
# variables to calculate triangular numbers
n = 0
p = 1
# while loop
while True:
# test if triangular number is a square number
if math.sqrt(n) % 1 == 0:
print int(math.sqrt(n))
# calculate triangular number
n += p
p += 1
```

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