A228826 Delayed continued fraction of sqrt(2).

2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1, -2, 1, 2, -1

Offset

0,1

Comments

See A228825 for a definition of delayed continued fraction (DCF).

DCF(r) is periodic if and only if CF(r) is periodic; DCF(sqrt(n)) is shown here for selected values of n,using Mathematica notation for periodic continued fractions.

n ........ DCF(sqrt(n))

2 ........ {2, {-1,-2,1,2}}

3 ........ {{1,2,-1,-1,-2,1}}

5 ........ {3, {-2,2,-1,-2,2,-2,1,2}}

6 ........ {3, {-1,-2,2,-2,1,2}}

7 ........ {2, {1,1,2,-2,2,-1,-1,-1,-1,-2,2,-2,1,1}}

8 ........ {2, {2,-2,2,-1,-1,-2,2,-2,1,1}}

10........ {4, {-2,2,-2,2,-1,-2,2,-2,2,-2,1,2}}

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,-1).

Formula

From Colin Barker, Sep 13 2013: (Start)

a(n) = ((2-i)*(-i)^n + (2+i)*i^n)/2 where i=sqrt(-1).

a(n) = -a(n-2).

G.f.: (2-x)/(x^2+1). (End)

Example

convergents: 2, 1, 4/3, 3/2, 10/7, 7/5, 24/17, 17/12, 58/41, 41/29, 140/99, ...

Mathematica

$MaxExtraPrecision = Infinity; x[0] = Sqrt[2]; s[x_] := s[x] = If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]; a[n_] := a[n] = s[Abs[x[n]]]*Sign[x[n]]; x[n_] := 1/(x[n - 1] - a[n - 1]); t = Table[a[n], {n, 0, 100}]
LinearRecurrence[{0, -1}, {2, -1}, 50] (* G. C. Greubel, Aug 19 2018 *)

Prog

(PARI) Vec(-(x-2)/(x^2+1) + O(x^100)) \\ Colin Barker, Sep 13 2013
(Magma) I:=[2, -1]; [n le 2 select I[n] else  - Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 19 2018

Crossrefs

Cf. A133570, A228825

Sequence in context: A167964 A280193 A327767 * A288699 A168361 A107393

Adjacent sequences: A228823 A228824 A228825 * A228827 A228828 A228829

Keyword

cofr,sign,easy

Author

Clark Kimberling, Sep 04 2013

Status

approved