Using the following information:
- The EROI of corn ethanol is 1.3:1
- Gross system energy per unit out = $EROI / [EROI-1]$
- This means for each extra unit of energy produced, 4.33 units of energy were produced, 3.33 of which were used to produce the extra unit
- You can only grow corn once per year
How many years do you have to grow corn to double your energy? Would that be 3.33 years?
EROI is energy output divided by energy input, which I understand to be 1.3 in this case.
Suppose you start with one unit of energy. Note that $1.3^0 = 1$.
After one cycle of growing corn and making ethanol etc., your energy will be $1.3 \cdot 1 = 1.3 = 1.3^1$.
After two cycles, your energy will be $1.3 \cdot 1.3 = 1.3^2 = 1.69$.
After three cycles, you will have $1.3 \cdot 1.3^2 = 1.3^3 = 2.197 > 2$.
The general solution is that after $n$ cycles (where $n$ is an integer) you will have $1.3^n$ times the original energy. To solve when the energy has grown to $k$ times the original, you need to solve the equation $1.3^n = k$. Take a logarithm of both sides to get $$ n = \frac{\log k}{\log 1.3} = \log_{1.3} k. $$