A259592 Denominators of the other-side convergents to sqrt(3).
1, 2, 4, 7, 15, 26, 56, 97, 209, 362, 780, 1351, 2911, 5042, 10864, 18817, 40545, 70226, 151316, 262087, 564719, 978122, 2107560, 3650401, 7865521, 13623482, 29354524, 50843527, 109552575, 189750626, 408855776, 708158977, 1525870529, 2642885282, 5694626340
Offset: 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
- Clark Kimberling, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).
Crossrefs
Programs
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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}] u = Denominator[t] -
PARI
Vec(-(x+1)*(x^2-x-1)/(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) = Q(i).
a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-x-1) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015
a(n) = 3^(n/2 + 1/2 - t)*((2 + sqrt(3))^t - (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - Ridouane Oudra, Aug 03 2021
Comments