Let r≥4 be a positive intger. Let us consider the difference non-autonomous equation:
u_{n+1}=(1+r^{2n+1})u_{n}-r^{2n-1}u_{n-1}+2 \tag......................{*}
All solutions of (*) have the form:
u_{n}=\sum_{m=1}^{n}\left(2(m-1)+u_1-ru_0\right)r^{n^2-m^2}+u_0r^{n^2}.....(**)
where u_0,u_1 are the initial conditions. All solutions of (*) tends toward infinity.
I have the followind question:
1) Find sufficient and necessary conditions in which u_{n} is an integer.
2) Is this claim correct:
Assume that there exist positive integers n, i,j,k and two initial conditions u_0,u_1 such that:
u_{n-1}=i
u_{n}=j
u_{n+1}=k...............(***)
Then there exist infinitely many n verifying similar equalities to (***).
See untitled1.pdf for the equations.
Dear Omar
I am aware of that book. In any case it has no relation with the question. Thank you.
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