# Time:  O(1)
# Space: O(1)

class Solution(object):
    def maxA(self, N):
        """
        :type N: int
        :rtype: int
        """
        if N <= 6: return N
        if N == 10: return 20

        n  = N // 5 + 1  # n3 + n4 increases one every 5 keys 
        # (1) n     =     n3 +     n4
        # (2) N + 1 = 4 * n3 + 5 * n4
        #     5 x (1) - (2) => 5*n - N - 1 = n3
        n3 = 5*n - N - 1  
        n4 = n - n3
        return 3**n3 * 4**n4


# Time:  O(n)
# Space: O(1)
class Solution2(object):
    def maxA(self, N):
        """
        :type N: int
        :rtype: int
        """
        if N < 7:
            return N
        dp = [i for i in xrange(N+1)]
        for i in xrange(7, N+1):
            dp[i % 6] = max(dp[(i-4) % 6]*3, dp[(i-5) % 6]*4)
        return dp[N % 6]
 

# Time:  O(1)
# Space: O(1)

class Solution(object):
    def maxA(self, N):
        """
        :type N: int
        :rtype: int
        """
        if N < 7: return N
        if N == 10: return 20

        n  = N // 5 + 1  # n3 + n4 increases one every 5 keys 
        # (1) n     =     n3 +     n4
        # (2) N + 1 = 4 * n3 + 5 * n4
        #     5 x (1) - (2) => 5*n - N - 1 = n3
        n3 = 5*n - N - 1  
        n4 = n - n3
        return 3**n3 * 4**n4


# Time:  O(n)
# Space: O(1)
class Solution2(object):
    def maxA(self, N):
        """
        :type N: int
        :rtype: int
        """
        if N < 7:
            return N
        dp = [i for i in xrange(N+1)]
        for i in xrange(7, N+1):
            dp[i % 6] = max(dp[(i-4) % 6]*3, dp[(i-5) % 6]*4)
        return dp[N % 6]
 

Update 4-keys-keyboard.py