old_contents: # Time:  O(n)
  # Space: O(1)
  
  class Solution(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]
   
  
new_contents: # 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]
   
  
subject: Update 4-keys-keyboard.py