Pythagorean triples

Do you ever need to come up with an unusual Pythagorean triple (a set of 3 natural numbers such that a² + b² = c²) quickly?  If you're not a math teacher, maybe you don't.

Here's what I use:

3,4,5 and multiples (6,8,10; 9,16,20; ...)

5,12,13 and multiples

7, 24, 25 and multiples

8,15,17 and multiples

But then my memory gives out, so I have to revert to a formula. For any natural number p, you get

2p, p² - 1, p² + 1.

So for p = 8  you get 16,63,65.

I wondered what other algebraic tricks I could find, and came upon

4p, p² - 4, p² + 4.

So for p = 5 I got 20, 21, 29. Since 29 is prime I know the triple isn't just a multiple of an earlier one.  Odd p values work well with this one:

p = 7: 28, 45, 53

p = 9: 36, 77, 85

Where can we go from here?

Open Letter to Demetri Martin

Dear Demetri,

As a fan of your work I humbly offer the following for your consideration.  In your "Large Pad" routine, you refer to a "simple chart" relating three variables: how short someone is, how drunk he is, and how funny it is.  You provide a 2-dimensional graph when a 3D graph is required to show the amount of funniness for each combination of shortness and drunkness:




The graph you provided appeared to depict shortness as [exponentially] dependent on drunkness, which is not what you intended.

I hope this clarifies the relationship you seek to communicate.

Sincerely,

Nathan Shields