writing a number as a sum of odd integers

Question

How many ways are there of writing $n$ as a sum of odd integers, where the order doesn't matter? For example, there are $2$ ways of writing $3$: $(1,1,1)$ and $(3)$.

Answer 1

A list of the first 10 cases to study:

1- 1 (1)

2- 1 (1,1)

3- 2 (1,1,1);(3)

4- 2 (1,1,1,1);(3,1)

5- 3 (1,1,1,1,1);(3,1,1);(5)

6- 4 (1,1,1,1,1,1);(3,1,1,1);(3,3);(5,1)

7- 5 (1,1,1,1,1,1,1);(3,1,1,1,1);(3,3,1);(5,1,1);(7)

8- 6 (1,1,1,1,1,1,1,1);(3,1,1,1,1,1);(3,3,1,1);(5,1,1,1);(5,3);(7,1)

9- 8 (1,1,1,1,1,1,1,1,1);(3,1,1,1,1,1,1);(3,3,1,1,1);(3,3,3);(5,1,1,1,1);(5,3,1);(7,1,1);(9)

10- 10 (1,1,1,1,1,1,1,1,1,1);(3,1,1,1,1,1,1,1);(3,3,1,1,1,1);(3,3,3,1);(5,1,1,1,1,1);(5,3,1,1);(5,5);(7,1,1,1);(7,3);(9,1)

One pattern I have noticed is that when n=${2^x}+1$, the first difference of the combination amounts goes up by 1. For example, starting when n=$5$, the combination number goes up steadily by ones. Then when n=$9$, it begins to go up steadily by twos.