045.py

#!/usr/bin/python
# -*- coding: utf-8 -*-

#Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
#Triangle 	  	Tn=n(n+1)/2 	  	1, 3, 6, 10, 15, ...
#Pentagonal 	  	Pn=n(3n−1)/2 	  	1, 5, 12, 22, 35, ...
#Hexagonal 	  	Hn=n(2n−1) 	  	1, 6, 15, 28, 45, ...

#It can be verified that T285 = P165 = H143 = 40755.

#Find the next triangle number that is also pentagonal and hexagonal.

#Answer:
	#1533776805

from time import time; t=time()
from mathplus import isqrt

n = 143
while True:
    n += 1
    hn = n*(2*n-1)
    vpn = hn*6
    vn = isqrt(vpn)+1
    if vn % 3 == 0 and vn * (vn-1) == vpn:
        print(hn)#, time()-t
        break

# http://mathworld.wolfram.com/HexagonalPentagonalNumber.html