Tuesday, March 12, 2013

The Math Blog: Pentagonal Numbers

The Math Blog is a blog just to talk about math and nothing else. Okay, toleration of moderate off-topic stuff is allowed, but don't get too off topic or else I will take that away with comment moderation.

Anyway, I have planned this:

Year 1 (2013): Number Theory & Awesome Mathematical Patterns
Year 2 (2014): Algebra

Pentagonal Numbers
Pentagonal numbers are numbers governed by the formula:

p_n = \tfrac{3n^2-n}{2} (credit to Wikipedia)

A visual representation is at this link:

http://en.wikipedia.org/wiki/File:Pentagonal_number.gif

Pentagonal numbers are awesome, and comparing them to triangular & square numbers:

p_n = \tfrac{3n^2-n}{2} <= Pentagonal
T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +n = \frac{n(n+1)}{2} = {n+1 \choose 2} <= Triangular (credit to Wikipedia)
n * n <= Square

Quite amazing.

Next up, hexagonal numbers!
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