r"""
Python module for OEIS sequence number A000172.

Franel number a(n) = Sum C(n,k)^3, k=0..n.

(n+1)^2 * a(n+1) = (7n^2+7n+2) * a(n) + 8n^2 * a(n-1).

Examples of use.
-----------------------------------------------------------------------
>>> from a000172 import *
>>> print a000172_list(10)
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162, 5280932]
>>> print a000172_offset
0
>>> for x in a000172_list_pairs(6):
...     print x
...
(0, 1)
(1, 2)
(2, 10)
(3, 56)
(4, 346)
(5, 2252)
>>> a000172_list_upto(10**6)
[1, 2, 10, 56, 346, 2252, 15184, 104960, 739162]
>>> print a000172(3)
56
-----------------------------------------------------------------------
"""

from itertools import islice, izip, takewhile, count

__all__ = ('a000172_offset', 'a000172_list', 'a000172_list_pairs', 'a000172_list_upto', 'a000172', 'a000172_gen')
__author__ = 'Nick Hobson <nickh@qbyte.org>'

a000172_offset = offset = 0

def a000172_gen():
    """Generator function for OEIS sequence A000172."""
    x, y = 1, 2
    for n in count(2):
        yield x
        x, y = y, ((7*n**2 - 7*n + 2)*y + (8*(n-1)**2)*x) / n**2

def a000172_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(a000172_gen(), n))

def a000172_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), a000172_gen()))

def a000172_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, a000172_gen()))

def a000172(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(a000172_gen(), n-offset, n-offset+1)).pop()