If I have a sequence, $\{t_1, \,t_2, \,t_1+t_2,\, t_1+2t_2,\cdots\}$ I know that the formula for it is:
$T_1=t_1, T_2=t_2, T_{n+2}=T_{n+1}+T_n.$
If there are two sequences, $A_n, B_n$ such that $A_1=1, A_2=0, A_{n+1}+A_n =A_{n+2}, B_1=0, B_2=1, B_{n+2}=B_{n+1}+B_n$
Then $A_n=F_{n-1}$ and $ B_n=F_{n-1}$ with $F_n$ being the Fibonacci numbers and $F_1=1, F_2=1$.
So the sequence becomes $T_n =t_1 F_{n-2} +t_2F_{n-1}$
The characteristic equation, which comes from $T_{n+2}=T_{n+1}+T_{n}$ is $x^2-x-1=0$ which has the roots $\frac{1\pm \sqrt{5}}{2}$. This is the same characteristic equation as the Fibonacci sequence.
Is it possible for two different sequences to have the same characteristic equation?