Years ago (1999), my drum buddy Warren Ashford asked the question- how many ways are there to partition n beats where order counts, but not using any 1's, the idea being that people don't hear a 1 alone, but as part of one of its neighbouring parts. He noticed a connection with Fibonacci numbers.

For example, here are the 1,1,2,3,5,8 ways to divide, respectively, 2,3,4,5,6 beats into parts as described.

2

3

4, 2+2

5, 2+3, 3+2

6, 2+4, 3+3, 4+2, 2+2+2

7, 2+5, 3+4, 4+3, 2+2+3, 5+2, 2+3+2, 3+2+2

One can easily prove the connection to Fibonacci numbers inductively, by considering the partitions that end in a 2 or not. The former are obtained by adding a 2 to a partition of n-2, and the latter can be obtained by adding a 1 to n-1 and absorbing the 1 into the last part, thus making the last part greater than 2.