I want to find the number of integer compositions of 19 using numbers 2 and 3.
If I wanted to find the number of integer compositions of 19 using 1 and 2, I could write it as $F(n)=F(n-1)+F(n-2)$ which would hint towards the Fibonacci Series.
Now, I am curious if I could write a similar equation for this example, and say, $F(n)=F(n-2)+F(n-3)$ which could hint towards a series that I could manipulate? However, I cannot come up with anything.
I also know that there is a solution that makes use of generating functions, but I do not really know how to do it for this specific example.
I would appreciate any help/hints!
It's essentially the same as the 1s and 2s problem. There's one composition of 1 made with 1s and 2s (namely 1 itself) and two compositions of 2 made with 1s and 2s (2 and 11), then the Fibonacci recurrence $F(n) = F(n-1) + F(n-2)$ leads you to the answer for 19.
For compositions with 2s and 3s, the initial values are 0, 1, 1 (no compositions of 1, one each for 2 and 3) and that's enough for the recurrence $P(n) = P(n-2) + P(n-3)$ you mentioned to kick in. The first few terms of the sequence are 0, 1, 1, 1, 2, 2, 3, 4, 5, 7. These are sometimes called the Padovan numbers.