OFFSET
0,2
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) = Q(i).
EXAMPLE
For r = e, the first 13 other-side convergents are 3/1, 5/2, 11/4, 19/7, 30/11, 106/39, 193/71, 299/110, 1457/536, 2721/1001, 4178/1537, 25946/9545, 49171/18089.
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 2/1 < e < 3/1 -1
1 3/1 > e > 5/2 1
2 8/3 < e < 11/4 -1
3 11/4 > e > 19/7 1
4 19/7 < e < 30/11 -1
5 87/32 > e > 106/39 1
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Jul 17 2015
STATUS
approved