extranapkins's sequence in functional Python

Inspired by Douglas Hofstadter’s and Martin Gardner’s deeply profound works on mathematical sequences and recursion, and Joel Grus’s talk Stupid Itertools Tricks for Data Science, I present a functional Python implementation of extranapkins’s sequence.1

The 0th item in this sequence is

"Fuck off"

The 1st item is

"I've got two words for you: Fuck off"

The 2nd item is

"I've got eight words for you: I've got two words for you: Fuck off"

The n+1th item in this sequence is

"I've got L_n words for you: S_n"

where S_n is the nth item in the sequence, and L_n is the number of words in S_n, expressed in words. Thus, here is a Python function that takes the nth item and returns the n+1th item in extranapkins’s sequence:2

from num2words import num2words

def next_ens(s):
    n = num2words(len(s.split()))
    return "I've got {} words for you: {}".format(n, s)

Given next_ens() we can generate successive items in the sequence by hand:

>>> next_ens('Fuck off')
"I've got two words for you: Fuck off"
>>> next_ens(next_ens('Fuck off'))
"I've got eight words for you: I've got two words for you: Fuck off"

We could automate this with recursion or a loop. But we can do it more quickly, with less code, and at the same time eliminate the dangers of variable mutation, the irritant of sys.setrecursionlimit, and the inefficiencies of recursion with memoization, by using the tools of functional programming.

In particular we’ll use Joel Grus’s python implementation of Haskell’s iterate() function. This takes a function f and an initial value x, and returns an iterator that lazily yields the items in the infinite sequence x, f(x), f(f(x)), f(f(f(x))), …

from itertools import accumulate, repeat

def iterate(f, x):
    return accumulate(repeat(x), lambda fx, _: f(fx))

You might have to stare at iterate() for a while (or watch Joel’s talk) to figure out how it does what it does. I also needed a pen and paper. Having to stare at a line for ten minutes is part of the hazing ritual of functional programming. After a while you can stop caring how these functions work, and take them on trust.

Anyway, we can use next_ens(), which tells you how to get from the nth to the n+1th item, and iterate(), to create an iterator ens that yields the items in the sequence…

>>> ens = iterate(next_ens, 'Fuck off')
>>> next(ens)
'Fuck off'
>>> next(ens)
"I've got two words for you: Fuck off"
>>> next(ens)
"I've got eight words for you: I've got two words for you: Fuck off"

…and with Joel’s Python implementation of take()

from itertools import islice

def take(n, it):
    return [x for x in islice(it, n)]

…we can evaluate the first n items…

>>> ens = iterate(next_ens, ‘Fuck off’)
>>> take(5, ens)
[‘Fuck off’, “I’ve got two words for you: Fuck off”, “I’ve got eight words for you: I’ve got two words for you: Fuck off”, “I’ve got fourteen words for you: I’ve got eight words for you: I’ve got two words for you: Fuck off”, “I’ve got twenty words for you: I’ve got fourteen words for you: I’ve got eight words for you: I’ve got two words for you: Fuck off”]

…or just the nth item:

>>> ens = iterate(next_ens, ‘Fuck off’)
>>> take(100, ens)[-1]
“I’ve got eight hundred and thirty-one words for you: I’ve got eight hundred and twenty-two words for you: I’ve got eight hundred and thirteen words for you: I’ve got eight hundred and four words for you: I’ve got seven hundred and ninety-five words for you: I’ve got seven hundred and eighty-six words for you: I’ve got seven hundred and seventy-seven words for you: I’ve got seven hundred and sixty-eight words for you: I’ve got seven hundred and fifty-nine words for you: I’ve got seven hundred and fifty words for you: I’ve got seven hundred and forty-one words for you: I’ve got seven hundred and thirty-two words for you: I’ve got seven hundred and twenty-three words for you: I’ve got seven hundred and fourteen words for you: I’ve got seven hundred and five words for you: I’ve got six hundred and ninety-six words for you: I’ve got six hundred and eighty-seven words for you: I’ve got six hundred and seventy-eight words for you: I’ve got six hundred and sixty-nine words for you: I’ve got six hundred and sixty words for you: I’ve got six hundred and fifty-one words for you: I’ve got six hundred and forty-two words for you: I’ve got six hundred and thirty-three words for you: I’ve got six hundred and twenty-four words for you: I’ve got six hundred and fifteen words for you: I’ve got six hundred and six words for you: I’ve got five hundred and ninety-seven words for you: I’ve got five hundred and eighty-eight words for you: I’ve got five hundred and seventy-nine words for you: I’ve got five hundred and seventy words for you: I’ve got five hundred and sixty-one words for you: I’ve got five hundred and fifty-two words for you: I’ve got five hundred and forty-three words for you: I’ve got five hundred and thirty-four words for you: I’ve got five hundred and twenty-five words for you: I’ve got five hundred and sixteen words for you: I’ve got five hundred and seven words for you: I’ve got five hundred words for you: I’ve got four hundred and ninety-one words for you: I’ve got four hundred and eighty-two words for you: I’ve got four hundred and seventy-three words for you: I’ve got four hundred and sixty-four words for you: I’ve got four hundred and fifty-five words for you: I’ve got four hundred and forty-six words for you: I’ve got four hundred and thirty-seven words for you: I’ve got four hundred and twenty-eight words for you: I’ve got four hundred and nineteen words for you: I’ve got four hundred and ten words for you: I’ve got four hundred and one words for you: I’ve got three hundred and ninety-two words for you: I’ve got three hundred and eighty-three words for you: I’ve got three hundred and seventy-four words for you: I’ve got three hundred and sixty-five words for you: I’ve got three hundred and fifty-six words for you: I’ve got three hundred and forty-seven words for you: I’ve got three hundred and thirty-eight words for you: I’ve got three hundred and twenty-nine words for you: I’ve got three hundred and twenty words for you: I’ve got three hundred and eleven words for you: I’ve got three hundred and two words for you: I’ve got two hundred and ninety-three words for you: I’ve got two hundred and eighty-four words for you: I’ve got two hundred and seventy-five words for you: I’ve got two hundred and sixty-six words for you: I’ve got two hundred and fifty-seven words for you: I’ve got two hundred and forty-eight words for you: I’ve got two hundred and thirty-nine words for you: I’ve got two hundred and thirty words for you: I’ve got two hundred and twenty-one words for you: I’ve got two hundred and twelve words for you: I’ve got two hundred and three words for you: I’ve got one hundred and ninety-four words for you: I’ve got one hundred and eighty-five words for you: I’ve got one hundred and seventy-six words for you: I’ve got one hundred and sixty-seven words for you: I’ve got one hundred and fifty-eight words for you: I’ve got one hundred and forty-nine words for you: I’ve got one hundred and forty words for you: I’ve got one hundred and thirty-one words for you: I’ve got one hundred and twenty-two words for you: I’ve got one hundred and thirteen words for you: I’ve got one hundred and four words for you: I’ve got ninety-eight words for you: I’ve got ninety-two words for you: I’ve got eighty-six words for you: I’ve got eighty words for you: I’ve got seventy-four words for you: I’ve got sixty-eight words for you: I’ve got sixty-two words for you: I’ve got fifty-six words for you: I’ve got fifty words for you: I’ve got forty-four words for you: I’ve got thirty-eight words for you: I’ve got thirty-two words for you: I’ve got twenty-six words for you: I’ve got twenty words for you: I’ve got fourteen words for you: I’ve got eight words for you: I’ve got two words for you: Fuck off”

So there you go: extranapkins’s sequence in functional Python.


  1. Seriously though, Joel’s talk is really good. It demonstrates that iterate is useful for much more than a recursively defined sequence with an F-bomb at the end. It can be used in any situation where an operation is applied repeatedly, such as gradient descent, and can lead to much simpler code. [return]
  2. This function uses num2words to convert an integer such as 129 into its English language words ("one hundred and twenty-nine"). Note that this package supports several languages, included Indian English and Latvian, so it would be simple to generalize extranapkins’s sequence to those languages. [return]