project_euler.problem_045.sol1

Problem 45: https://projecteuler.net/problem=45

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 * (3 * n - 1)) / 2 1, 5, 12, 22, 35, … Hexagonal H(n) = n * (2 * n - 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. All triangle numbers are hexagonal numbers. T(2n-1) = n * (2 * n - 1) = H(n) So we shall check only for hexagonal numbers which are also pentagonal.

Functions

hexagonal_num(→ int)	Returns nth hexagonal number
is_pentagonal(→ bool)	Returns True if n is pentagonal, False otherwise.
solution(→ int)	Returns the next number which is triangular, pentagonal and hexagonal.

Module Contents

project_euler.problem_045.sol1.hexagonal_num(n: int) → int

Returns nth hexagonal number >>> hexagonal_num(143) 40755 >>> hexagonal_num(21) 861 >>> hexagonal_num(10) 190

project_euler.problem_045.sol1.is_pentagonal(n: int) → bool

Returns True if n is pentagonal, False otherwise. >>> is_pentagonal(330) True >>> is_pentagonal(7683) False >>> is_pentagonal(2380) True

project_euler.problem_045.sol1.solution(start: int = 144) → int

Returns the next number which is triangular, pentagonal and hexagonal. >>> solution(144) 1533776805