# nayuki/Project-Euler-solutions

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 # # Solution to Project Euler problem 78 # Copyright (c) Project Nayuki. All rights reserved. # # https://www.nayuki.io/page/project-euler-solutions # https://github.com/nayuki/Project-Euler-solutions # import itertools MODULUS = 10**6 def compute(): partitions = [1] for i in itertools.count(len(partitions)): # We calculate partitions[i] mod 10^6 using a formula based on generalized pentagonal numbers: # partitions(i) = partitions(i - pentagonal(1)) + partitions(i - pentagonal(-1)) # - partitions(i - pentagonal(2)) - partitions(i - pentagonal(-2)) # + partitions(i - pentagonal(3)) + partitions(i - pentagonal(-3)) # - partitions(i - pentagonal(4)) - partitions(i - pentagonal(-4)) # + ..., # where pentagonal(j) = (3*n^2 - n) / 2, and # we stop the sum when i - pentagonal(+/-j) < 0. # Note that for j > 0, pentagonal(j) < pentagonal(-j) < pentagonal(j+1). # # (The formula is used without mathematical justification; # see https://en.wikipedia.org/wiki/Partition_(number_theory)#Generating_function .) item = 0 for j in itertools.count(1): sign = -1 if j % 2 == 0 else +1 index = (j * j * 3 - j) // 2 if index > i: break item += partitions[i - index] * sign index += j # index == (j * j * 3 + j) // 2 if index > i: break item += partitions[i - index] * sign item %= MODULUS # Check or memoize the number if item == 0: return str(i) partitions.append(item) if __name__ == "__main__": print(compute())