How many recurrence relations are possible for a sequence?

Question

How many recurrence relations are possible for a sequence? Example: $$ 5, 11, 29, 83, 245, \ldots $$ We have two recurrence relation:

1. $T_n = 3T_{n-1} - 4$

2. $T_n = T_{n-1} + 6 \cdot 3^{n-1}$

Both give $T_n = 3^n + 2$

Answer 1

Let $s$ be an arbitrary number. Then the recurrence $$T_n=s\left(3T_{n-1}-4\right)+(1-s)\left( T_{n-1}+6\cdot 3^{n-1} \right)$$ gives the same sequence for any $s$.