We found a recursive formula for the following problem:
For any positive integer $n$, let $b(n)$ be the number of ways that you can write $n$ as a sum using only the numbers 1, 2, and 3 where the order of the sum doesn’t matter.
our recursive formula is $$b(n) = b(n-3)+b(n-2)-b(n-5)+1$$ from the values $$b(0)=1, b(1)=1, b(2)=2, b(3)=3, b(4)=4, b(5)=5, b(6)=7, b(7)=8, b(8)=10, b(9)=12, b(10)=14$$
we are trying to find a non-recursive formula for this sequence, so that we can find the values for $b$ at 2012 through 2017
please also explain HOW to get the formula, as this is for marks.