A259594 Denominators of the other-side convergents to sqrt(6).
1, 3, 11, 29, 109, 287, 1079, 2841, 10681, 28123, 105731, 278389, 1046629, 2755767, 10360559, 27279281, 102558961, 270037043, 1015229051, 2673091149, 10049731549, 26460874447, 99482086439, 261935653321, 984771132841, 2592895658763, 9748229241971
Offset: 0
Examples
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
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).
Programs
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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*) -
PARI
Vec(-(x+1)*(x^2-2*x-1)/(x^4-10*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) = 10*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-2*x-1) / (x^4-10*x^2+1). - Colin Barker, Jul 21 2015
Comments