Building a tower using colorful blocks How many possibilities are there to build a tower of n height, using colorful blocks, where:


*

*white block is of 1 height

*black, red, blue, green, yellow, pink blocks are equal to 2 heights


I need to find the generating function formula for this.
So, for n = 1 I get 1 possibility, 
for n = 2 I get 2 possibilities, 
for n = 3 I get 3 possibilities
for n = 4 I get > 4 possibilities etc.
The generating function at this moment would be $1 + 2x + 3x^{2} + ...$. But I have no idea how can I find the general formula calculating this.
Could you give me any suggestions, or solutions (;-)) ?

Let the number of towers of height $n$ be $a_n$. To build a tower of height $n$, you start with a tower of height $n - 1$ and add a white block ($a_{n - 1}$ ways) or with a tower of height $n - 2$ and add one of 6 height-2 blocks. In all, you can write:
$$
a_{n + 2} = a_{n + 1} + 6 a_n
$$
Clearly $a_0 = a_1 = 1$.
Define $A(z) = \sum_{n \ge 0} a_n z^n$, multiply the recurrence by $z^n$ and sum over $n \ge 0$ to get:
$$
\frac{A(z) - a_0 - a_1 z}{z^2} = \frac{A(z) - a_0}{z} + 6 A(z)
$$
You get:
$$
A(z) = \frac{1}{1 - z - 6 z^2}
     = \frac{3}{5} \cdot \frac{1}{1 - 3 z} + \frac{2}{5} \cdot \frac{1}{1 + 2 z}
$$
This is just geometric series:
$$
a_n = \frac{3}{5} \cdot 3^n + \frac{2}{3} \cdot (-2)^n
    = \frac{3^{n + 1} - (-2)^{n + 1}}{5}
$$
