OFFSET
0,1
COMMENTS
Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.
FORMULA
p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
EXAMPLE
For r = Pi, the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532.
A comparison of convergents with other-side convergents:
i p(i)/q(i) P(i)/Q(i) p(i)*Q(i) - P(i)*q(i)
0 3/1 < Pi < 4/1 -1
1 22/7 > Pi > 25/8 1
2 333/106 < Pi < 355/113 -1
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Jul 17 2015
STATUS
approved