Question: Let $a_{n}$ count the number of different ways to build a tower $n$ units high using red 1-unit blocks, red 2-unit blocks, blue 1-unit blocks, and blue 3-unit blocks. Find $a_{n}$ for $n \geq 1$.
So I have found a recurrence relation for this, namely $a_{n} = 2a_{n-1} + a_{n-2} + a_{n-3}$.
But I have also modelled this using exponential generating functions given by $g(x) = \left( 1 + x + \frac{x^{2}}{2!}+ \dots \right)^{2}(1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \dots)(1 + \frac{x^{3}}{3!} + \frac{x^{6}}{6!} + \dots )$
When I compute the coefficient of $\frac{x^3}{3!}$, they agree with those given by my recurrence relation, up until $a_{3}$, where it gives $a_{3} = 15$ (although it should be 13). Is there some reason why my generating function would not model the above relation?