# Infinite sequences

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Generators can be used to represent infinite sequences:

``````def integers_starting_from(n):
while True:
yield n
n += 1

natural_numbers = integers_starting_from(1)``````

Infinite sequence of numbers as above can also be generated with the help of `itertools.count`. The above code could be written as below

``natural_numbers = itertools.count(1)``

You can use generator comprehensions on infinite generators to produce new generators:

``````multiples_of_two = (x * 2 for x in natural_numbers)
multiples_of_three = (x for x in natural_numbers if x % 3 == 0)``````

Be aware that an infinite generator does not have an end, so passing it to any function that will attempt to consume the generator entirely will have dire consequences:

``list(multiples_of_two)  # will never terminate, or raise an OS-specific error``

Instead, use list/set comprehensions with `range` (or `xrange` for python < 3.0):

``````first_five_multiples_of_three = [next(multiples_of_three) for _ in range(5)]
# [3, 6, 9, 12, 15]``````

or use `itertools.islice()` to slice the iterator to a subset:

``````from itertools import islice
multiples_of_four = (x * 4 for x in integers_starting_from(1))
first_five_multiples_of_four = list(islice(multiples_of_four, 5))
# [4, 8, 12, 16, 20]``````

Note that the original generator is updated too, just like all other generators coming from the same “root”:

``````next(natural_numbers)    # yields 16
next(multiples_of_two)   # yields 34
next(multiples_of_four)  # yields 24``````

An infinite sequence can also be iterated with a `for`-loop. Make sure to include a conditional `break` statement so that the loop would terminate eventually:

``````for idx, number in enumerate(multiplies_of_two):
print(number)
if idx == 9:
break  # stop after taking the first 10 multiplies of two``````

# Classic example - Fibonacci numbers

``````import itertools

def fibonacci():
a, b = 1, 1
while True:
yield a
a, b = b, a + b

first_ten_fibs = list(itertools.islice(fibonacci(), 10))
# [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]

def nth_fib(n):
return next(itertools.islice(fibonacci(), n - 1, n))

ninety_nineth_fib = nth_fib(99)  # 354224848179261915075``````