See above, that's fairly close to what my code does.

Pardon me for rewriting in Python. My code runs without recursion and at about the same speed. def StairCaseMemo ( N ): memo = {} def stepsM ( N ): if N == 0 : return 1 elif N< 0 : return 0 if N in memo.keys(): return memo[N] else : memo[N] = stepsM(N - 1 ) + stepsM(N - 2 ) + stepsM(N - 3 ); return memo[N]; print(stepsM(N)) def StairCaseMine ( limit ): # initial setup array = {} array[ 0 ] = 1 array[ 1 ] = array[ 2 ] = index = 0 while index < limit: count = array[index] array[index+ 1 ] += count array[index+ 2 ] += count array[index+ 3 ] = count index += 1 print(array[limit]) Enjoy!