This is a concept that I vaguely understand, but I'd like to get an intuitive understanding of how to write a recurrence relation of the form: $$ t_{n}-3t_{n-1}+2t_{n-2}=0 $$ subject to $$ t_0=2, t_1=3 $$ in terms of the shift operator $E$, in order to derive a characteristic polynomial. I already know this particular equation can be expressed as: $$ (E^{2}-3E+2)t=0 $$ I understand that the second expression is defined by the operations done on a solution sequence $t$ such that $t_{1...n}=0$, but is there an easier way to come up with that expression that simply writing the first few terms of the sequence and moving things around?
Edit: fixed second expression (+2, not -2)
(Your shift equation should be $(E^2-3E+2)t=0$: you have the wrong sign on the constant term.)
You can read it straight from the recurrence: choose $n$ to make the smallest index $0$, and replace $at_k$ by $aE^k$ throughout.