def e(n):
"Return the nth convergent of the continued fraction for e."
if n == 1:
return 2
# Collect the first n-1 partial values of e.
values = deque(take(n-1, partial_values()))
# Construct the continued fraction, where 'tail' is the recursive component.
return Fraction(2 + Fraction(1, tail(values)))
def convergent(D, n):
"""Return the nth convergent of the continued fraction for sqrt(D),
where D is a non-square positive integer."""
if n == 1:
return next(process_cf(D))
# Collect the first n partial values of D.
values = deque(take(n, process_cf(D)))
# Construct the continued fraction, where 'tail' is the recursive component.
return Fraction(values.popleft() + Fraction(1, tail(values)))
def p(n):
if n < 0:
return 0
elif n == 0:
return 1
else:
# Generating pentagonals is repeated many times, should think about optimising this.
pentagonals = list(takewhile(lambda x: x <= n, generalised_pentagonals()))
terms = [p(n - x) for x in pentagonals]
coefs = list(take(len(terms), cycle([1, 1, -1, -1])))
return sum(a*b for (a, b) in zip(terms, coefs))
def is_lychrel(n):
start = n + rev(n)
iterations = iterate(lambda x: x + rev(x), start)
return not any(is_palindromic(y) for y in take(50, iterations))
def problem57():
generate_tails = iterate(tail, 2) # yields 2, tail(2), tail(tail(2)), ...
expansions = (1 + Fraction(1, t) for t in generate_tails)
return quantify(take(1000, expansions), pred=check_numerator)