The Math Blog: March 2013

Saturday, March 16, 2013

The Math Blog: Centered Square Numbers

The Math Blog is back with centered square numbers!

In elementary number theory, a centered square number is like a number with a dot in a middle, then making dots around it making rotated squares of side length adding 2 every time. Wikipedia provides a useful diagram at this link.

Centered square numbers are represented like this: C_{4, n.}

The formula for finding a the nth centered square number is:

Bye! $C_{4,n} = {(2n-1)^2 + 1 \over 2};$

Friday, March 15, 2013

The Math Blog: Octagonal Numbers

Headline: $n^2 + 4\sum_{k = 1}^{n - 1} k = 3n^2-2n$

I might as well say that this blog is currently managed along with some other blogs, which I'll mention later.

Octagonal numbers grow like pentagonal, hexagonal, and heptagonal numbers. The formula for any octagonal number n is:

$O_n = 3n^2-2n$ (credit to Wikipedia)

Bye for now!

MATH NEWS IS NOW BEING AWESOME WITH FEATURED SEQUENCE:

3, 7, 31, 127...

Guess what the sequence is! Answer in the next post.

Thursday, March 14, 2013

The Math Blog: Hexagonal Numbers

$h_n= 2n^2-n = n(2n-1) = {{2n}\times{(2n-1)}\over 2}.\,\!$

I will do the Math Blog every day, unless I am too busy. If I am too busy, then I will post twice on a day.

Today is awesome math with hexagonal numbers! Hexagonal numbers are also pretty awesome, and this is the formula for deriving hexagonal numbers:

$h_n= 2n^2-n = n(2n-1) = {{2n}\times{(2n-1)}\over 2}.\,\!$ (again, credit to Wikipedia)

All hexagonal numbers are triangular numbers, but only every other triangular number is a hexagonal numbers. Hexagonal numbers can be diagrammatically be represented at this link here.

From this, you can derive the first hexagonal numbers:

1, 6, 15, 28, 45...

Every even perfect number is hexagonal, as 6 and 28 are hexagonal and so is 496: you can solve this equation on your own using algebra. I will solve it because I feel like it. If you don't want to watch an algebraic proof, skip this section.

2n^2 - n - 496 = 0

I used a calculator at http://www.webgraphing.com/quadraticequation_factoring.jsp, FYI. The perfect number 496 is the 16th hexagonal number.

Try it yourself; 512 - 16 = 496.

In fact, no odd perfect number is found yet, so it comes to reasoning that all perfect numbers that we have found are hexagonal, and yet, that statement is CORRECT!

Bye!

~Albert Tam

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!