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A linear nonhomogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form: $ a_n= c_1 a_{n-1}+c_2 a_{n-2}+\ldots+c_ka_{n-k}+f(n)$ where $c_1,c_2,\ldots,c_k$ are real numbers and $f(n)$ is a function not identically zero depending only on n.

I wonder that if $f(n)$ is an arbitrary function then all recurrent relation is a linear nonhomogeneous recurrence relation? For example, if $a_n=a_{n-1} +a_{n-2}^2$ we can definite $f(n)=a_{n-2}^2$

T.Nguyen
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  • $a_{n-2}$ does not depend only on $n$, but also on $a_{n-3}$ etc They mean something like $a_n=2a_{n-1}+n^2+n+1$ where $f(n)=n^2+n+1$ depends only on $n$. They are quite difficult to solve, BTW – Raffaele Jul 15 '17 at 17:46
  • you can think of a sequence as a function $\mathbb{N} \rightarrow \mathbb{R}$ . So $f(n)= a_{n-2}^2$ is a function which depends only on n. – T.Nguyen Jul 15 '17 at 17:56

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