Is this sequence a recurrence relation? $21, 36, 55, 60, 67, 68, 92, 93, 125$
I thought maybe it's $T_{2n + 4} + 2(n + 4)$ and I tried a few other formulas involving triangular numbers. I also tried variations on the Fibonacci sequence, like adding $2$ to some terms. This riddle has me stumped.
 A: For integer sequences, you can always use oeis.org (Online Encyclopedia for Integer Sequences). To answer your question: It's not a recurrence relation. Spoilers if you click the link.

 It's actually opus numbers for Beethoven's nine symphonies.

http://oeis.org/search?q=21%2C36%2C55%2C60%2C67%2C68%2C92%2C93%2C125&language=english&go=Search
A: Anytime I don't know what a sequence of integers is (or sometimes sequences of rational numbers), I turn to N. J. A. Sloane's On-Line Encyclopedia of Integer Sequences (OEIS).
If you search this particular sequence, you should get Sloane's A001491, the opus numbers of Beethoven's nine symphonies. Maybe you think Mozart is a better composer, maybe you just like Dvořák better, but neither of them can say all his symphonies were published in his lifetime in pretty much the order he wrote them. Beethoven can say that (if you ignore the Tenth Symphony).
These opus numbers don't have a neat mathematical explanation, but rather are the result of various circumstances of Beethoven's life. For example, notice that he worked on the Fifth and Sixth Symphonies at about the same time, and I think they were even premiered in the same concert on the same day. The Seventh and the Eighth also came out at almost the same time, and kind of make a trilogy with Wellingtons Sieg, which is sometimes referred to as a symphony.
Still, I think it's an interesting coincidence that the opus numbers of his first three symphonies happen to be second hexagonal numbers (numbers of the form $2n^2 + n$). That sequence goes: 21, 36, 55, 78, 105, 136, 171, 210, 253, ... Is it any wonder then that Sloane wrote this:

This was a mystery for many years - even Clifford Pickover could not recall the explanation. It was finally identified in 1998 by Derek Holt.

Lastly, I would like to mention there are some people who see the Fibonacci numbers in Beethoven's Diabelli Variations. It's possible that Beethoven knew about the golden section... 
A: You could always find the polynomial that take on these values for $1, 2, 3, \ldots, 9$ and claim you've found a sequence, but when oeis doesn't know it (for anything mathematical), it's most likely not interesting.
