# 3n+2
def rule(x):
    if x == 2:
        return True
    return rule(x-3) if x > 2 else False
"""
2 \in S

n \in S
_______
(n+3) \in S
"""

# 2n+3m+1
def rule2(x):
    if x == 1:
        return True
    return rule2(x-2) or rule2(x-3) if x > 1 else False

"""
1 \in S

n \in S
_________
(n+2) \in S

m \in S
_________
(m+3) \in S
"""

# (n, 2n+1)
def rule3(x):
    if x == (0, 1):
        return True
    return rule3((x[0]-1, x[1]-2)) if x[0] > 0 else False

"""
(0, 1) \in S

(n, m) \in S
____________
(n+1, m+1) \in S
"""

# (n, n*n)
def rule4(x):
    if x == (0, 0):
        return True
    return rule4((x[0]-1, x[1]-2*x[0]+1)) if x[0] > 0 else False

"""
(0, 0) \in S

(n, m) \in S
____________
(n+1, m+2*n+1) \in S
"""

______________________________________

1.2

"""
(0, 1) \in S

(n, k) \in S
____________
(n+1, k+7) \in S
"""
# (n, 7n+1) -> (0, 1), (1, 8), (2, 15), (3, 22)...

"""
(0, 1) \in S

(n, k) \in S
____________
(n+1, 2k) \in S
"""
# (n, 2^n) \in S -> (0, 1), (1, 2), (2, 4), (3, 8)

"""
(0, 0, 1) \in S

(n, i, j) \in S
_______________
(n+1, j, i+j) \in S
"""
# fibonacci

"""
(0, 1, 0) \in S

(n, i, j) \in S
_______________
(n+1, i+2, i+j) \in S
"""
# (n, 2n+1, n^2)

______________________________________

1.3

"""
0 \in S
1 \in S

n \in S
_______
(n+3) \in S
"""