r"""
Python module for OEIS sequence number A005374.

a(0) = 0; for n > 0, a(n) = n - a(a(a(n-1))).

Examples of use.
-----------------------------------------------------------------------
>>> from a005374 import *
>>> print a005374_list(18)
[0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12]
>>> print a005374_offset
0
>>> for x in a005374_list_pairs(6):
...     print x
...
(0, 0)
(1, 1)
(2, 1)
(3, 2)
(4, 3)
(5, 4)
>>> a005374_list_upto(9)
[0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9]
>>> print a005374(200)
136
>>> a005374_list_inv(136)
[199, 200]
-----------------------------------------------------------------------
"""

from itertools import islice, izip, takewhile, count

__all__ = ('a005374_offset', 'a005374_list', 'a005374_list_pairs',
           'a005374_list_upto', 'a005374', 'a005374_gen', 'a005374_list_inv')
__author__ = 'Nick Hobson <nickh@qbyte.org>'

a005374_offset = offset = 0

def a005374_gen():
    """Generator function for OEIS sequence A005374."""
    a = {0:0}
    yield a[0]
    for n in count(1):
        a[n] = n - a[a[a[n-1]]]
        yield a[n]

def a005374_list(n):
    """Returns a list of the first n >= 0 terms."""
    if n < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(islice(a005374_gen(), n))

def a005374_list_pairs(n):
    """Returns a list of tuples (n, a(n)) of the first n >= 0 terms."""
    if n < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(izip(xrange(offset, n+offset), a005374_gen()))

def a005374_list_upto(m):
    """Returns a list of all terms not exceeding m >= 0."""
    if m < 0: raise ValueError, 'Input must be a non-negative integer'
    return list(takewhile(lambda t: t <= m, a005374_gen()))

def a005374(n):
    """Returns the term with index n >= 0; offset 0."""
    if n < offset: raise ValueError, 'Input must be an integer >= offset = ' + str(offset)
    return list(islice(a005374_gen(), n-offset, n-offset+1)).pop()

def a005374_list_inv(m):
    """Returns a list of all indices n for which a(n) = m >= 0."""
    if m < 0: raise ValueError, 'Input must be a non-negative integer'
    r, n = [], 0
    for x in a005374_gen():
        if x > m: break
        if x == m: r.append(n)
        n += 1
    return r