According to my book, a linear homogeneous recurrence of order $k$ is expressed this way: $A_0a_n+A_1a_{n-1}+A_2a_{n-2}+\cdots+A_ka_{n-k}=0$ While a linear non-homogeneous recurrence of order $k$ is this way: $A_0a_n+A_1a_{n-1}+A_2a_{n-2}+\cdots+A_ka_{n-k}=f(n)$
I hardly understand what that is supposed to mean. There is not much explanation. At first, I thought that linear homogeneous were equalities to $0$ while linear non-homogeneous were equalities to something else.
Well, I was wrong because later the book says that the succession defined by $c_n=c_{n-1}+4c_{n-3}$ is linear homogeneous of order 3.
Can you give me a better explanation?