Polygonal Number Generating Function and Formula

Today my diurnal age is 25233, a number that happens to be a polygonal number, specifically a 32-gonal number. A WolframAlpha article states that the generating function for the n-gonal numbers is given by: $$G_n(x)= \frac{x \, [(n-3)x+1]}{(1-x)^3} $$ This means that for the 32-gonal numbers, the formula becomes: $$G_{32}(x)= \frac{x \, (33x+1)}{(1-x)^3}$$In WolframAlpha, the coefficients can then be identified using the series command:

In Sage, the Taylor series is generated using the code:

g(x)=x*(33*x+1)/(1-x)^3
g.taylor(x,0,15).coefficients()

When run, this code produces the following output:

In the case of the 36-gonal numbers, the formula for the n-th term is given by:$$a(n)=n(17n-16) $$The general formula for the n-th polygonal number is given by:$$a(n)=(P-2) \frac{n(n+1)}{2}-(P-3)n$$where \(P\) is the number of vertices of the polygon.

In the case of \(P=36 \) (our 36-gon), the formula becomes:$$34\frac{n(n+1)}{2}-33n=n(17n-16) $$Here's a video describing how the formula is derived:

Taylor Series

For a really superb video explaining how Taylor Polynomials and Taylor Series are created, you need go no further than this video by 3Blue1Brown.

$$ \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} =f(a)+\frac {f'(a)}{1!} (x-a)+ \frac{f''(a)}{2!} (x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+ \cdots $$The formula above was copied from Wikipedia where all mathematical expressions are in LaTeX but enclosed within <math> ... </math> tags. While it's always good practice to create the LaTeX from scratch, it's still useful to remember that any mathematical expressions found in Wikipedia can be copied directly into Blogger.

Anyway, the main point of this post is to remind myself to watch more of 3Blue1Brown's videos. The Taylor Series video was created in May of 2017 and since then twenty more have been added. Altogether he has almost seventy videos of which I've watched sixteen. The description of the channel states that:

3Blue1Brown, by Grant Sanderson, is some combination of math and entertainment, depending on your disposition. The goal is for explanations to be driven by animations and for difficult problems to be made simple with changes in perspective. 

He can be followed on Twitter at https://twitter.com/3blue1brown and a website at http://www.3blue1brown.com.