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the-number-of-ways-to-make-the-sum.py
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# Time: O(1)
# Space: O(1)
# math
class Solution(object):
def numberOfWays(self, n):
"""
:type n: int
:rtype: int
"""
MOD = 10**9+7
# n = 1
# 1
# n = 2
# extend:
# 1 1
# new:
# 2 (extra)
# n = 3
# extend:
# 1 1 1
# 1 2
# n = 4
# extend:
# 1 1 1 1
# 1 1 2
# new:
# 2 2
# 4 (extra)
# n = 5
# extend
# 1 1 1 1 1
# 1 1 1 2
# 1 2 2
# 1 4
# n = 6
# extend:
# 1 1 1 1 1 1
# 1 1 1 1 2
# 1 1 2 2
# 1 1 4
# new:
# 2 2 2
# 6 (extra)
# 2 4
# n = 7
# extend
# 1 1 1 1 1 1 1
# 1 1 1 1 1 2
# 1 1 1 2 2
# 1 1 1 4
# 1 2 2 2
# 1 6
# 1 2 4
# n = 8
# extend:
# 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 2
# 1 1 1 1 2 2
# 1 1 1 1 4
# 1 1 2 2 2
# 1 1 6
# 1 1 2 4
# new:
# 2 2 2 2
# 2 6 (1x2, 0x4)
# 2 2 4 (2x2, 1x4)
# 4 4 (extra, 0x2, 2x4)
# n = 9
# extend:
# 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 2
# 1 1 1 1 1 2 2
# 1 1 1 1 1 4
# 1 1 1 2 2 2
# 1 1 1 6
# 1 1 1 2 4
# 1 2 2 2 2
# 1 2 6 (1x2, 0x4)
# 1 2 2 4 (2x2, 1x4)
# 1 4 4 (0x2, 2x4)
# n = 10
# extend:
# 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 2
# 1 1 1 1 1 1 2 2
# 1 1 1 1 1 1 4
# 1 1 1 1 2 2 2
# 1 1 1 1 6
# 1 1 1 1 2 4
# 1 1 2 2 2 2
# 1 1 2 6
# 1 1 2 2 4
# 1 1 4 4
# new:
# 2 2 2 2 2
# 2 2 2 4
# 2 2 6 (2x2, 0x4)
# 6 4 (extra, 0x2, 1x4)
# 2 4 4 (1x2, 2x4)
# n = 11
# extend:
# 1 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 2
# 1 1 1 1 1 1 1 2 2
# 1 1 1 1 1 1 1 4
# 1 1 1 1 1 2 2 2
# 1 1 1 1 1 6
# 1 1 1 1 1 2 4
# 1 1 1 2 2 2 2
# 1 1 1 2 6
# 1 1 1 2 2 4
# 1 1 1 4 4
# 1 2 2 2 2 2
# 1 2 2 2 4
# 1 2 2 6 (2x2, 0x4)
# 1 6 4 (0x2, 1x4)
# 1 2 4 4 (1x2, 2x4)
# n = 12
# extend:
# 1 1 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1 2
# 1 1 1 1 1 1 1 1 2 2
# 1 1 1 1 1 1 1 1 4
# 1 1 1 1 1 1 2 2 2
# 1 1 1 1 1 1 6
# 1 1 1 1 1 1 2 4
# 1 1 1 1 2 2 2 2
# 1 1 1 1 2 6
# 1 1 1 1 2 2 4
# 1 1 1 1 4 4
# 1 1 2 2 2 2 2
# 1 1 2 2 2 4
# 1 1 2 2 6
# 1 1 6 4
# 1 1 2 4 4
# new:
# 2 2 2 2 2 2
# 2 2 2 2 4
# 2 2 2 6
# 6 6 (extra, 0x2, 0x4)
# 2 6 4 (1x2, 1x4)
# 2 2 4 4 (2x2, 2x4)
# n = 13
# extend:
# 1 1 1 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1 1 2
# 1 1 1 1 1 1 1 1 1 2 2
# 1 1 1 1 1 1 1 1 1 4
# 1 1 1 1 1 1 1 2 2 2
# 1 1 1 1 1 1 1 6
# 1 1 1 1 1 1 1 2 4
# 1 1 1 1 1 2 2 2 2
# 1 1 1 1 1 2 6
# 1 1 1 1 1 2 2 4
# 1 1 1 1 1 4 4
# 1 1 1 2 2 2 2 2
# 1 1 1 2 2 2 4
# 1 1 1 2 2 6
# 1 1 1 6 4
# 1 1 1 2 4 4
# 1 2 2 2 2 2 2
# 1 2 2 2 2 4
# 1 2 2 2 6
# 1 6 6 (0x2, 0x4)
# 1 2 6 4 (1x2, 1x4)
# 1 2 2 4 4 (2x2, 2x4)
# => f(n) = 1+sum(i//2 if i%2 == 0 else 0 for i in xrange(2, n+1))
return (1+((n//2)+1)*(n//2)//2)%MOD
# Time: O(1)
# Space: O(1)
# math
class Solution2(object):
def numberOfWays(self, n):
"""
:type n: int
:rtype: int
"""
MOD = 10**9+7
def count_1_2_6(n):
# sum((n-6*i)//2+1 for i in xrange((n//6)+1))
# = sum(((n//2)+1)-3*i for i in xrange((n//6)+1))
return (n//2+1)*((n//6)-0+1)-3*((n//6)+0)*((n//6)-0+1)//2
return reduce(lambda x, y: (x+count_1_2_6(n-4*y))%MOD, (i for i in xrange(min(n//4, 2)+1)), 0)
# Time: O(n)
# Space: O(1)
# math
class Solution3(object):
def numberOfWays(self, n):
"""
:type n: int
:rtype: int
"""
MOD = 10**9+7
def count_1_2(n):
return n//2+1
def count_1_2_6(n):
return sum(count_1_2(n-6*i) for i in xrange((n//6)+1))
return reduce(lambda x, y: (x+count_1_2_6(n-4*y))%MOD, (i for i in xrange(min(n//4, 2)+1)), 0)
# Time: O(n)
# Space: O(n)
# dp
class Solution4(object):
def numberOfWays(self, n):
"""
:type n: int
:rtype: int
"""
MOD = 10**9+7
dp = [0]*(n+1)
dp[0] = 1
for i in (1, 2, 6):
for j in xrange(i, n+1):
dp[j] += dp[j-i]
return reduce(lambda x, y: (x+dp[n-4*y])%MOD, (i for i in xrange(min(n//4, 2)+1)), 0)