#!/usr/bin/env python3

# Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
#
# Triangle	 	T_n=n(n+1)/2	1, 3, 6, 10, 15, ...
# Pentagonal	P_n=n(3n−1)/2	1, 5, 12, 22, 35, ...
# Hexagonal	 	H_n=n(2n−1)	 	1, 6, 15, 28, 45, ...
#
# It can be verified that T_285 = P_165 = H_143 = 40755.
#
# Find the next triangle number that is also pentagonal and hexagonal.

from projecteuler import is_pentagonal, timing


@timing
def p045() -> None:
    found = 0
    i = 143

    while not found:
        i = i + 1
        # Hexagonal numbers are also triangle numbers, so it's sufficient
        # to generate hexagonal numbers (starting from H_144) and check if
        # they're also pentagonal.
        n = i * (2 * i - 1)

        if is_pentagonal(n):
            found = 1

    print('Project Euler, Problem 45')
    print(f'Answer: {n}')


if __name__ == '__main__':
    p045()