Problem 45 : Triangular, pentagonal, and hexagonal

Problem Statement

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 T₂₈₅ = P₁₆₅ = H₁₄₃ = 40755.

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

Solution

def pentagonal(n):
    if (1+(24*n+1)**0.5) % 6 == 0:
        return True
    return False

def hexagonal(n):
    if (1+(8*n+1)**0.5)%4 == 0:
        return True
    return False


flag =True

while flag:
    for i in range(1,10**6):
        a=int(i*(i+1)/2)
        if hexagonal(a) and pentagonal(a):
            if a > 40755: 
                print(a)
                flag=False
                break

Output

1533776805