SnakeWay checker

 def dfs(l, st, f, k, callback):
     l[st[0]][st[1]] = k
     if st == f and k == len(l) * 3:
         callback(l)
         l[st[0]][st[1]] = None
         return
     candidates = (
     (st[0] - 1, st[1]),
     (st[0] + 1, st[1]),
     (st[0], st[1] - 1),
     (st[0], st[1] + 1))
     for c in candidates:
         if c[0] < 0 or c[0] >= len(l) or c[1] < 0 or c[1] > 2 or l[c[0]][c[1]]:
             continue
         dfs(l, c, f, k+1, callback)
     l[st[0]][st[1]] = None

def check(n):
    cnt = 0
    l = [[None] * 3 for _ in range(n)]
    def c(_):
        nonlocal cnt
        cnt += 1
    dfs(l, (0, 0), (n - 1, 0), 1, c)
    cnt_l = cnt
    cnt = 0
    dfs(l, (0, 0), (n - 1, 2), 1, c)
    cnt_d = cnt
    cnt = 0
    dfs(l, (0, 0), (0, 2), 1, c)
    cnt_s = cnt
    return cnt_d, cnt_l, cnt_s

for i in range(1, 11):
    print(check(i))
# Out:
# (1, 0, 1)
# (1, 1, 0)
# (2, 2, 2)
# (4, 4, 0)
# (8, 8, 4)
# (16, 16, 0)
# (32, 32, 8)
# (64, 64, 0)
# (128, 128, 16)
# (256, 256, 0)