How many ways are there to add up odd integers to 20? How many ways are there to add up odd integers to 20?
Here, $1+19$ is one solution, $19+1$ is a different solution, and $1+1+\dots+1$ counts as just one solution. 
 A: To sum up to $20$, we could:


*

*first write a $1$ and count all the ways to add up to the remainder, $19$, or

*first write a $3$ and count all the ways to add up to the remainder, $17$, or

*first write a $5$ and count all the ways to add up to the remainder, $15$, or …


The answer will be the sum of all these counts.

Let’s denote the number of lists of odd positive integers that sum to $n$ as $a_n$.
We can easily see that $a_1 = 1$ (just $1$) and $a_2 = 1$ (just $1+1$).
We can generalize our observation and say that, for $n \geq 1$,
$$a_n = a_{n-1} + a_{n-3} + a_{n-5} + \dots \tag{i}$$
ending at either $a_1$ (if $n$ is even) or $a_0$ (if $n$ is odd).
But then also for $n \geq 3$,
$$ a_{n-2} = a_{n-3} + a_{n-5} + \dots \tag{ii}$$
so combining $\text{(i)}$ and $\text{(ii)}$ we find
$$ a_n = a_{n-1} + a_{n-2}, $$
meaning $a_n$ are just the Fibonacci numbers ($1, 1, 2, 3, 5, \dots$).
Thus $a_{20}$ is the twentieth Fibonacci number, $6765$.
A: Length[Flatten[Permutations /@ IntegerPartitions[20, All, 2 Range[10] - 1], 1]]  
6765 ways, according to Mathematica.
