Patience, young "Padovan"
Jelly, 10 bytes

9s3’Ẓæ*³FṀ


Try it online!

1-indexed. Computes the largest element of: $$\begin{bmatrix}0&amp;0&amp;1 \\ 1&amp;0&amp;1 \\ 0&amp;1&amp;0\end{bmatrix}^n$$
where the binary matrix is conveniently computed as: $$\begin{bmatrix}\mathsf{isprime}(0)&amp;\mathsf{isprime}(1)&amp;\mathsf{isprime}(2) \\ \mathsf{isprime}(3)&amp;\mathsf{isprime}(4)&amp;\mathsf{isprime}(5) \\ \mathsf{isprime}(6)&amp;\mathsf{isprime}(7)&amp;\mathsf{isprime}(8)\end{bmatrix}$$

(this is a total coincidence.)

9s3         [[1,2,3],[4,5,6],[7,8,9]]    9 split 3
   ’        [[0,1,2],[3,4,5],[6,7,8]]    decrease
    Ẓ       [[0,0,1],[1,0,1],[0,1,0]]    isprime
     æ*³    [[0,0,1],[1,0,1],[0,1,0]]^n  matrix power by input
        FṀ                               flatten, maximum