Domino and Tromino - Transitions

From profile $00$, you can: place a vertical domino (stay at $00$), place two horizontal dominoes stacked (stay at $00$), place a tromino in row $0$ (move to $01$), or place a tromino in row $1$ (move to $10$).

From profile $01$, you can: place a horizontal domino in row $1$ (move to $00$), or place a tromino covering row $1$'s protruding cell and extending row $0$ (move to $10$).

From $10$, transitions are symmetric. These define formulas: let $f[i]$ be ways with profile $00$, $g[i]$ for $01$, $h[i]$ for $10$. Then $f[i+1] = 2f[i] + g[i] + h[i]$, $g[i+1] = f[i] + h[i]$, $h[i+1] = f[i] + g[i]$.