RMO 2025 K4

A frog is initially at $(0, 0)$ and it reaches $(n, 2)$ , $n \ge 1$ , using the following moves in any order several times:

$R=(1, 0)$ , that is, if the frog is at $(a, b)$ it goes to $(a+1, b)$ ; $U=(0, 1)$ , that is, if the frog is at $(a, b)$ it goes to $(a, b+1)$ ; $D=(1, 1)$ , that is, if the frog is at $(a, b)$ it goes to $(a+1, b+1)$ .

In how many ways can the frog go from $(0, 0)$ to $(n, 2)$ , $n \ge 1$ , using the above steps subject to the condition that steps of the type $UU, DD$ are forbidden? (For example, for $n=3$ , $RDUR, DRD$ are admissible paths, while $DDR, RUURR$ are not.)