How many ways are there to additively partition a natural number?

partitions.py

# How many unique ways are there to add positive integers to get the number N?
# https://oeis.org/A000041

from itertools import count, chain, islice


def partitions():
    """
    >>> list(islice(partitions(), 20))
    [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490]
    """
    values = [1]
    while True:
        yield values[-1]
        next_value = 0
        for sign, offset in zip(signs(), offsets()):
            if offset > len(values):
                break
            next_value += sign * values[-offset]
        values.append(next_value)


def signs():
    """
    >>> list(islice(signs(), 8))
    [1, 1, -1, -1, 1, 1, -1, -1]
    """
    while True:
        for s in [+1, +1, -1, -1]:
            yield s


def offsets():
    """
    >>> list(islice(offsets(), 10))
    [1, 2, 5, 7, 12, 15, 22, 26, 35, 40]
    """
    offset = 1
    for diff in offset_diffs():
        yield offset
        offset += diff


def offset_diffs():
    """
    >>> list(islice(offset_diffs(), 10))
    [1, 3, 2, 5, 3, 7, 4, 9, 5, 11]
    """
    yield from chain.from_iterable(zip(count(1), count(3, step=2)))


if __name__ == '__main__':
    # https://www.youtube.com/watch?v=iJ8pnCO0nTY
    index = 665
    print(list(islice(partitions(), index, index+1))[0])