>>> a = [1, 2]
>>> type(a)
<type 'list'>
>>> type(iter(a))
<type 'listiterator'>
>>> it = iter(a)
>>> next(it)
1
>>> next(it)
2
>>> next(it)
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
StopIteration
>>> len(a)
2
>>> len(it)
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
TypeError: object of type 'listiterator' has no len()

class count_iterator(object):
  n = 0
 
  def __iter__(self):
    return self
 
  def next(self):
    y = self.n
    self.n += 1
    return y

>>> counter = count_iterator()
>>> next(counter)
0
>>> next(counter)
1
>>> next(counter)
2
>>> next(counter)
3
>>> list(counter) # This will result in an infinite loop!

class SimpleList(object):
  def __init__(self, *items):
    self.items = items
 
  def __getitem__(self, i):
    return self.items[i]

>>> a = SimpleList(1, 2, 3)
>>> it = iter(a)
>>> next(it)
1
>>> next(it)
2
>>> next(it)
3
>>> next(it)
Traceback (most recent call last):
 File "<stdin>", line 1, in <module>
StopIteration

Q(n)=Q(n-Q(n-1))+Q(n?Q(n?2))

class qsequence(object):
  def __init__(self, s):
    self.s = s[:]
 
  def next(self):
    try:
      q = self.s[-self.s[-1]] + self.s[-self.s[-2]]
      self.s.append(q)
      return q
    except IndexError:
      raise StopIteration()
 
  def __iter__(self):
    return self
 
  def current_state(self):
    return self.s

>>> Q = qsequence([1, 1])
>>> next(Q)
2
>>> next(Q)
3
>>> [next(Q) for __ in xrange(10)]
[3, 4, 5, 5, 6, 6, 6, 8, 8, 8]

def count_generator():
  n = 0
  while True:
   yield n
   n += 1

>>> counter = count_generator()
>>> counter
<generator object count_generator at 0x106bf1aa0>
>>> next(counter)
0
>>> next(counter)
1
>>> iter(counter)
<generator object count_generator at 0x106bf1aa0>
>>> iter(counter) is counter
True
>>> type(counter)
<type 'generator'>

def hofstadter_generator(s):
  a = s[:]
  while True:
    try:
      q = a[-a[-1]] + a[-a[-2]]
      a.append(q)
      yield q
    except IndexError:
      return

import random
 
def bernoulli_process(p):
  if p > 1.0 or p < 0.0:
    raise ValueError("p should be between 0.0 and 1.0.")
 
  while True:
    yield random.random() < p
 
def von_neumann_extractor(process):
  while True:
    x, y = process.next(), process.next()
    if x != y:
      yield x

>>> def tent_map(mu, x0):
...  x = x0
...  while True:
...    yield x
...    x = mu * min(x, 1.0 - x)
...
>>>
>>> t = tent_map(2.0, 0.1)
>>> for __ in xrange(30):
...  print t.next()
...
0.1
0.2
0.4
0.8
0.4
0.8
0.4
0.8
0.4
0.8
0.4
0.8
0.4
0.8
0.4
0.8
0.4
0.799999999999
0.400000000001
0.800000000003
0.399999999994
0.799999999988
0.400000000023
0.800000000047
0.399999999907
0.799999999814
0.400000000373
0.800000000745
0.39999999851
0.79999999702

def collatz(n):
  yield n
  while n != 1:
   n = n / 2 if n % 2 == 0 else 3 * n + 1
   yield n

>>> # If the Collatz conjecture is true then list(collatz(n)) for any n will
... # always terminate (though your machine might run out of memory first!)
>>> list(collatz(7))
[7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
>>> list(collatz(13))
[13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
>>> list(collatz(17))
[17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]
>>> list(collatz(19))
[19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

def permutations(items):
  if len(items) == 0:
    yield []
  else:
    pi = items[:]
    for i in xrange(len(pi)):
      pi[0], pi[i] = pi[i], pi[0]
      for p in permutations(pi[1:]):
        yield [pi[0]] + p

>>> for p in permutations([1, 2, 3]):
...   print p
...
[1, 2, 3]
[1, 3, 2]
[2, 1, 3]
[2, 3, 1]
[3, 1, 2]
[3, 2, 1]

>>> g = (x ** 2 for x in itertools.count(1))
>>> g
<generator object <genexpr> at 0x1029a5fa0>
>>> next(g)
1
>>> next(g)
4
>>> iter(g)
<generator object <genexpr> at 0x1029a5fa0>
>>> iter(g) is g
True
>>> [g.next() for __ in xrange(10)]
[9, 16, 25, 36, 49, 64, 81, 100, 121, 144]

>>> g = (random.random() < 0.4 for __ in itertools.count())
>>> [g.next() for __ in xrange(10)]
[False, False, False, True, True, False, True, False, False, True]

>>> sum(x ** 2 for x in xrange(10))
285

from itertools import combinations, product
 
n = 4
d = 3
 
def visit(*indices):
  print indices
 
# Loop through all possible indices of a 3-D array
for i in xrange(n):
  for j in xrange(n):
    for k in xrange(n):
      visit(i, j, k)
 
# Equivalent using itertools.product
for indices in product(*([xrange(n)] * d)):
  visit(*indices)
 
# Now loop through all indices 0 <= i < j < k <= n
for i in xrange(n):
  for j in xrange(i + 1, n):
    for k in xrange(j + 1, n):
      visit(i, j, k)
 
# And equivalent using itertools.combinations
for indices in combinations(xrange(n), d):
  visit(*indices)

import itertools
import math
 
def inversion_number(A):
  """Return the number of inversions in list A."""
  return sum(1 for x, y in itertools.combinations(xrange(len(A)), 2) if A[x] > A[y])
 
def total_inversions(n):
  """Return total number of inversions in permutations of n."""
  return sum(inversion_number(A) for A in itertools.permutations(xrange(n)))

>>> [total_inversions(n) for n in xrange(10)]
[0, 0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840]
 
>>> [math.factorial(n) * n * (n - 1) / 4 for n in xrange(10)]
[0, 0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840]

def count_fixed_points(p):
  """Return the number of fixed points of p as a permutation."""
  return sum(1 for x in p if p[x] == x)
 
def count_partial_derangements(n, k):
  """Returns the number of permutations of n with k fixed points."""
  return sum(1 for p in itertools.permutations(xrange(n)) if count_fixed_points(p) == k)

# Usage:
>>> [count_partial_derangements(6, i) for i in xrange(7)]
[265, 264, 135, 40, 15, 0, 1]