__all__ = ['sum_1_n', 'ltm', 'is_triangular']

import math

import scipy.special

def sum_1_n(n: int) -> int:

"""Calculates the sum from 1 to n.

see https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF

:rtype: int

:param n: the integer to sum up to

:return: sum from 1 to n

"""

assert n >= 0, 'n must not be negative!'

return int(n * (n + 1) / 2)

def ltm(n: int) -> int:

"""Returns the next lower triangular number to n or n if n is triangular.

:param n: the number to get the next lower triangular number too

:return: n if triangular else the next lower triangular number

:rtype: int

"""

assert n > 0, 'n must be greater than zero!'

n += 1

x1 = math.ceil((math.sqrt(8 * n + 1) - 1) /

2)

return int(scipy.special.binom(x1, 2))

def is_triangular(n: int) -> bool:

"""Checks if a positive integer is triangular.

:param n: the number to check

:return: True if n is triangular else False

:rtype: bool

"""

if n <= 0:

return False

return n == ltm(n)