### 51st Putnam 1990

**Problem A1**

Prove that the sequence a_{0} = 2, 3, 6, 14, 40, 152, 784, ... with general term a_{n} = (n+4) a_{n-1} - 4n a_{n-2} + (4n-8) a_{n-3} is the sum of two well-known sequences.

**Solution**

Answer: n! + 2^{n}.

*Easy.*

This is not a nice problem. We know the answer is easy (because A1 is almost always easy), so we are looking for something simple. Just try subtracting off various simple series until you recognize the result. I was lucky: 152, 784 vaguely reminded me of 120, 720.

51st Putnam 1990

© John Scholes

jscholes@kalva.demon.co.uk

3 Nov 1999