uzumaki

result

>>> from uzumaki import *
>>> [uzumaki(x) for x in range(20)]
[(0, 0), (1, 0), (1, 1), (0, 1), (-1, 1), (-1, 0), (-1, -1), (0, -1), (1, -1), (2, -1), (2, 0), (2, 1), (2, 2), (1, 2), (0, 2), (-1, 2), (-2, 2), (-2, 1), (-2, 0), (-2, -1)]
>>> uzumaki(1000000000)
(1683, -15811)

uzumaki.py

from math import sqrt, floor

def uzumaki(n):
    base = int(floor(sqrt(n)))
    delta = n-base**2
    if base%2 == 0:
        basex = -base/2
        basey = base/2
        xdir = 1
        ydir = -1
    else:
        basex = base/2+1
        basey = -base/2+1
        xdir = -1
        ydir = 1
    x = basex + xdir * (delta-base if delta>base else 0)
    y = basey + ydir * (delta if delta<base else base)
    return x,y