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

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Sierpinski TIrangle

Using the same method used in my previous Sierpinski Triangle program, which is written in Python, I wrote a fractal generator for my graphing calculator TI-84 Plus in BASIC.

Faintly visible Somewhat visible Strongly visible

"// TI-84 Plus BASIC Code"
"// Jonathan Frech 25th of April, 2016"
"//         edited 21st of May  , 2016"

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Pascal’s Triangle

Pascal’s triangle is an interesting mathematical sequence. It is often written as a triangle, starting with \{1\}, then \{1, 1\}. To generate the next row, you add the two numbers above to form another one. So the next row in the sequence is \{1, 2, 1\} then \{1, 3, 3, 1\}, \{1, 4, 6, 4, 1\} and so on (sequence A007318 in OEIS).

One interesting property of Pascal’s triangle is the generation of binomials.
To calculate (a + b)^4, you can look at the 4th row (listed above and when starting to count at 0) and determine
(a + b)^4 = (1 \cdot a^4 \cdot b^0) + (4 \cdot a^3 \cdot b^1) + (6 \cdot a^2 \cdot b^2) + (4 \cdot a^1 \cdot b^3) + (1 \cdot a^0 \cdot b^4)
(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.

This program generates Pascal’s sequence in a rather unusual shape, looking a bit like a crown.

To get more information about Pascal’s triangle, check out this Wikipedia entry.


  • F1 advances the sequence
  • Space takes a screenshot

Fontsize 50 Fontsize 10 Fontsize 1

# Python 2.7.7 Code
# Pygame 1.9.1 (for Python 2.7.7)
# Jonathan Frech 25th of March, 2016

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