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.
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).
FORMULA
p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
a(n) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: (x^3-3*x^2+7*x+3) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015
EXAMPLE
For r = sqrt(6), the first 7 other-side convergents are 3, 7/3, 27/11, 71/29, 267/109, 703/287, 2643/1079. 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 < sqrt(6) < 3/1 -1
1 5/2 > sqrt(6) > 7/3 1
2 22/9 < sqrt(6) < 27/11 -1
3 49/20 > sqrt(6) > 71/29 1
4 218/89 < sqrt(6) < 267/109 -1
5 485/198 > sqrt(6) > 703/287 1
MATHEMATICA
r = Sqrt[6]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t] (* A259594 *)
v = Numerator[t] (* A259595 *)
LinearRecurrence[{0, 10, 0, -1}, {3, 7, 27, 71}, 30] (* Harvey P. Dale, Mar 21 2023 *)
PROG
(PARI) Vec((x^3-3*x^2+7*x+3)/(x^4-10*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Clark Kimberling, Jul 20 2015
STATUS
approved