r"""
Python module for OEIS sequence number A000100.

a(n) = number of compositions of n in which the maximum part size is 3.
a(n) = 2*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4) - a(n-5).

Examples of use.
-----------------------------------------------------------------------
>>> from a000100 import *
>>> print a000100_list(15)
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360, 694, 1328, 2526]
>>> print a000100_offset
0
>>> for x in a000100_list_pairs(6):
...     print x
...
(0, 0)
(1, 0)
(2, 0)
(3, 1)
(4, 2)
(5, 5)
>>> a000100_list_upto(1000)
[0, 0, 0, 1, 2, 5, 11, 23, 47, 94, 185, 360, 694]
>>> print a000100(3)
1
-----------------------------------------------------------------------
"""

from itertools import islice, izip, takewhile, count

__all__ = ('a000100_offset', 'a000100_list', 'a000100_list_pairs', 'a000100_list_upto', 'a000100', 'a000100_gen')
__author__ = 'Nick Hobson <nickh@qbyte.org>'

a000100_offset = offset = 0

def a000100_gen():
    """Generator function for OEIS sequence A000100."""
    a = [0, 0, 0, 1, 2]
    for x in a:
        yield x
    for n in count(5):
        a[n%5] = 2*a[(n-1)%5] + a[(n-2)%5] - a[(n-3)%5] - 2*a[(n-4)%5] - a[(n-5)%5]
        yield a[n%5]

def a000100_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(a000100_gen(), n))

def a000100_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), a000100_gen()))

def a000100_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, a000100_gen()))

def a000100(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(a000100_gen(), n-offset, n-offset+1)).pop()