project_euler.problem_045.sol1
==============================

.. py:module:: project_euler.problem_045.sol1

.. autoapi-nested-parse::

   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
---------

.. autoapisummary::

   project_euler.problem_045.sol1.hexagonal_num
   project_euler.problem_045.sol1.is_pentagonal
   project_euler.problem_045.sol1.solution


Module Contents
---------------

.. py:function:: hexagonal_num(n: int) -> int

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


.. py:function:: 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


.. py:function:: solution(start: int = 144) -> int

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