A228825 -id:A228825 - cp's OEIS frontend

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

Clark Kimberling, Sep 04 2013

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, {-1,-2,1,2}}
........ {{1,2,-1,-1,-2,1}}
........ {3, {-2,2,-1,-2,2,-2,1,2}}
........ {3, {-1,-2,2,-2,1,2}}
........ {2, {1,1,2,-2,2,-1,-1,-1,-1,-2,2,-2,1,1}}
........ {2, {2,-2,2,-1,-1,-2,2,-2,1,1}}
....... {4, {-2,2,-2,2,-1,-2,2,-2,2,-2,1,2}}

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A133570, A228825

I:=[2,-1]; [n le 2 select I[n] else  - Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 19 2018

$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 *)

Vec(-(x-2)/(x^2+1) + O(x^100)) \\ Colin Barker, Sep 13 2013

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)

A280136 Negative continued fraction of e (or negative continued fraction expansion of e).

Original entry on oeis.org

3, 4, 3, 2, 2, 2, 3, 8, 3, 2, 2, 2, 2, 2, 2, 2, 3, 12, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 16, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 20, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 24, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Randy L. Ekl, Dec 26 2016

nonn

After the first term (3), a pattern of groups consisting, for m>=1, of the number 4m, followed by 3, then 4m-1 2's, then 3.

			e = 2.71828... = 3 - 1/(4 - 1/(3 - 1/(...))).

Leonard Eugene Dickson, History of the Theory of Numbers, page 379.

Cf. A003417 (continued fraction of e).
Cf. A005131 (generalized continued fraction of e).
Cf. A133570 (exact continued fraction of e).
Cf. A228825 (delayed continued fraction of e).
Cf. A280135 (negative continued fraction of Pi).

\p10000; p=exp(1.0); for(i=1, 300, print(i, " ", ceil(p)); p=ceil(p)-p; p=1/p )

More terms from Jinyuan Wang, Mar 04 2020

A228941 The n-th convergent of CF(e) is the a(n)-th convergent of DCF(e), the delayed continued fraction of e.

Original entry on oeis.org

1, 3, 4, 5, 9, 10, 11, 17, 18, 19, 27, 28, 29, 39, 40, 41, 53, 54, 55, 69, 70, 71, 87, 88, 89, 107, 108, 109

Offset: 1

Clark Kimberling, Sep 08 2013

See A228825 for a definition of delayed continued fraction. Is A014209 is a subsequence of A228941? It appears that the difference sequence of A228941, namely (2,1,1,4,1,1,6,1,1,...), is the continued fraction of (e-2)/(3-e).

			The convergents of CF(e) are 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, ...; the convergents of DCF(e) are 2, 5/2, 3, 8/3, 11/4, 30/11, 49/18, 68/25, 19/7, 87/32, 106/39,...; a(5) = 9 because 19/7 is the 9th convergent of DCF(e).

Cf. A228825, A014209.

$MaxExtraPrecision = Infinity; x[0] = E; s[x_] := s[x] = If[FractionalPart[x] < 1/2, Ceiling[x], Floor[x]]; f[n_] := f[n] = s[Abs[x[n]]]*Sign[x[n]]; x[n_] := 1/(x[n - 1] - f[n - 1]); t = Table[f[n], {n, 0, 120}] ;(* A228825; delayed cf of x[0] *); t1 = Convergents[t]; t2 = Convergents[ContinuedFraction[E, 120]]; Flatten[Table[Position[t1, t2[[n]]], {n, 1, 28}]]

Empirical g.f.: x*(x^5+x^3-x^2-2*x-1) / ((x-1)^3*(x^2+x+1)^2). - Colin Barker, Sep 13 2013

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A228826 Delayed continued fraction of sqrt(2).

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A280136 Negative continued fraction of e (or negative continued fraction expansion of e).

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A228941 The n-th convergent of CF(e) is the a(n)-th convergent of DCF(e), the delayed continued fraction of e.

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