L - probability of the left move
R - probability of the right move
S - probability of staying

After 1 move:

S

P_1 = S

After 2 moves:

SS

LR
RL

P_2 = S^2 + 2*LR


After 3 moves:

SSS

SLR
LSR
LRS

SRL
RSL
RLS

P_3 = S^3 + 2*3*LRS = S^3 + 2*?


After 4 moves:

SSSS

SSLR
SSRL
SLSR
SRSL
SLRS
SRLS

LSSR
RSSL
LSRS
RSLS
LRSS
RLSS

LRLR
LLRR
RLRL
RRLL

P_4 = S^4 + 12*LRS^2 + 4*L^2*R^2

After 5 moves:

SSSSS

.....