r"""
Python module for OEIS sequence number A126086.

Number of paths from (0,0,0) to (n,n,n) such that at each step:
(i) at least one coordinate increases,
(ii) no coordinate decreases,
(iii) no coordinate increases by more than 1, and
(iv) all coordinates are integers.

Examples of use.
-----------------------------------------------------------------------
>>> from a126086 import *
>>> print a126086_list(7)
[1, 13, 409, 16081, 699121, 32193253, 1538743249L]
>>> print a126086_offset
0
>>> for x in a126086_list_pairs(6):
...     print x
...
(0, 1)
(1, 13)
(2, 409)
(3, 16081)
(4, 699121)
(5, 32193253)
>>> a126086_list_upto(10**6)
[1, 13, 409, 16081, 699121]
>>> print a126086(5)
32193253
-----------------------------------------------------------------------
"""

from itertools import islice, izip, takewhile, count

__all__ = ('a126086_offset', 'a126086_list', 'a126086_list_pairs', 'a126086_list_upto', 'a126086', 'a126086_gen')
__author__ = 'Nick Hobson <nickh@qbyte.org>'

a126086_offset = offset = 0

def a126086_gen():
    """Generator function for OEIS sequence A126086."""
    a = [1, 13, 409, 16081]
    for x in a:
        yield x
    for n in count(4):
        a[n%4] = ((55*n**2 - 51*n + 9)*a[(n-1)%4] + 3*(37*n**2 - 150*n + 147)*a[(n-2)%4]
         - (n-3)*(5*n-12)*a[(n-3)%4] + (2*(n-3)**2)*a[(n-4)%4]) / n**2
        yield a[n%4]

def a126086_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(a126086_gen(), n))

def a126086_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), a126086_gen()))

def a126086_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, a126086_gen()))

def a126086(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(a126086_gen(), n-offset, n-offset+1)).pop()