LEFT AND RIGHT ONLY

2 states - 121 steps
1L00L1
0H01R0

3 states - 521 steps (found by Demonin)
1L00L1
1L21L0
0H00R0

4 states - 4386 steps (found by Demonin)
1L00L1
1L21L0
0R31R0
0H00R1

FORWARD AND U-TURNS ALLOWED

2 states - 121 steps
1L00L1
0H01R0

3 states - 21,790 steps
1R11F0
1R11R2
1H01F0

4 states - 287,251 steps
1F10U3
0R00F2
1H00U1
1F00F0