A259593 Numerators of the other-side convergents to sqrt(3).

2, 3, 7, 12, 26, 45, 97, 168, 362, 627, 1351, 2340, 5042, 8733, 18817, 32592, 70226, 121635, 262087, 453948, 978122, 1694157, 3650401, 6322680, 13623482, 23596563, 50843527, 88063572, 189750626, 328657725, 708158977, 1226567328, 2642885282, 4577611587

Offset: 0

Author

Clark Kimberling, Jul 20 2015

Keywords

• nonn
• easy
• frac

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.

Examples

			For r = sqrt(3), 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      1/1  < sqrt(3) <     2/1               -1
1      2/1  > sqrt(3) >     3/2                1
2      5/3  < sqrt(3) <     7/4               -1
3      7/4  > sqrt(3) >    12/7                1
4     19/11 < sqrt(3) <    26/15              -1
5     26/15 > sqrt(3) >    45/26               1
		

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Crossrefs

Cf. A002530, A002531, A259592 (denominators).

Cf. A001075 (even bisection), A005320 (odd bisection).

Programs

Mathematica

r = Sqrt[3]; 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}]
v = Numerator[t]

PARI

Vec(-(x^2-3*x-2)/(x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

Formula

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).

a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015

G.f.: -(x^2-3*x-2) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015

a(n) = 3^(n/2 - t + 1)*((2 + sqrt(3))^t + (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - Ridouane Oudra, Aug 03 2021