Note on a discrete initial value problem from a competition

Abstract

Abstract The following discrete initial value problem x n + 1 = x n ( x n − 1 2 − 2 ) − x 1 , n ∈ N , $$ x_{n+1}=x_{n}\bigl(x_{n-1}^{2}-2 \bigr)-x_{1},\quad n\in {\mathbb{N}}, $$ x 0 = 2 $x_{0}=2$ and x 1 = 5 / 2 $x_{1}=5/2$ , appeared at an international competition. It is known that the problem can be solved in closed form. Here we discuss the solvability of a more general initial value problem which includes the former one. We show that, in a sense, there are not so many solvable discrete initial value problems related to this one, showing its specificity, which is a bit surprising result.

Keywords

• Discrete initial value problem
• Nonlinear difference equation
• Closed-form formula
• Solvable equation