


from functools import lru_cache
@lru_cache
def 							UpperCamelCase_  (				__snake_case			:						int     )->      int:




						'''simple docstring'''
						if not isinstance(__snake_case	,			__snake_case     ):
												raise TypeError('k must be an integer.'     )
						if k < 0:
												raise ValueError('k integer must be greater or equal to zero.'     )
						if k == 0:
												return 0
						elif k == 1:
												return 1
						else:
												return fibonacci(k - 1     ) + fibonacci(k - 2     )
def 							UpperCamelCase_  (				__snake_case			:						list	,			__snake_case			:						int     )->      int:




						'''simple docstring'''
						__magic_name__ 		=			len(__snake_case     )
						# Find m such that F_m >= n where F_i is the i_th fibonacci number.
						__magic_name__ 		=			0
						while True:
												if fibonacci(__snake_case     ) >= len_list:
																		__magic_name__ 		=			i
																		break
												i += 1
						__magic_name__ 		=			0
						while fibb_k > 0:
												__magic_name__ 		=			min(
												    offset + fibonacci(fibb_k - 1     )	,			len_list - 1     )  # Prevent out of range
												__magic_name__ 		=			arr[index_k]
												if item_k_1 == val:
																		return index_k
												elif val < item_k_1:
																		fibb_k -= 1
												elif val > item_k_1:
																		offset += fibonacci(fibb_k - 1     )
																		fibb_k -= 2
						else:
												return -1


if __name__ == "__main__":
					import doctest

					doctest.testmod()





def 							UpperCamelCase_  (				__snake_case			:						int     )->      bool:




						'''simple docstring'''
						if not isinstance(__snake_case	,			__snake_case     ):
												__magic_name__ 		=			F'''Input value of [number={number}] must be an integer'''
												raise TypeError(__snake_case     )
						if number < 0 or number % 2 == 1:
												return False
						__magic_name__ 		=			int(number**0.5     )
						return number == number_sqrt * (number_sqrt + 1)


if __name__ == "__main__":
					import doctest

					doctest.testmod()

