A228667 -id:A228667 - cp's OEIS frontend

A228825 Delayed continued fraction of e.

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

Offset: 0

Clark Kimberling, Sep 04 2013

An algorithm for the (usual) continued fraction of r > 0 follows:  x(0) = r, a(n) = floor(x(n)), x(n+1) = 1/(x(n) - a(n)).
The accelerated continued fraction uses "round" instead of "floor" (cf. A133593, A133570, A228667), where round(x) is the integer nearest x.
The delayed continued fraction (DCF) uses "second nearest integer", so that all the terms are in {-2, -1, 1, 2}.  If s/t and u/v are consecutive convergents of a DCF, then |s*x-u*t| = 1.
Regarding DCF(e), after the initial (2,2), the strings (-1,-1,-1) and (1,1,1) alternate with odd-length strings of the forms (-2,2,...,-2) and (2,-2,...,2).  The string lengths form the sequence 2,3,3,3,5,3,7,3,9,3,11,3,13,3,...
Comparison of convergence rates is indicated by the following approximate values of x-e, where x is the 20th convergent: for delayed CF, x-e = 5.4x10^-7; for classical CF, x-e = 6.1x10^-16; for accelerated CF, x-e = -6.6x10^-27.  The convergents for accelerated CF are a proper subset of those for classical CF, which are a proper subset of those for delayed CF (which are sampled in Example).

			Convergents: 2, 5/2, 3, 8/3, 11/4, 30/11, 49/18, 68/25, 19/7, 87/32, 106/39, 299/110, 492/181,...

Cf. A133570, A228826

$MaxExtraPrecision = Infinity; x[0] = E; 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}]

A228668 Array: row n consists of n-th nonsquare f(n) followed by L(CF(sqrt(f(n)))) followed by L(ACF(sqrt(f(n)))), where L indicates the length of the repeating string; CF indicates continued fraction, and ACF indicates accelerated continued fraction.

Original entry on oeis.org

2, 1, 1, 3, 2, 2, 5, 1, 1, 6, 2, 2, 7, 2, 4, 8, 2, 2, 10, 1, 1, 11, 2, 2, 12, 2, 2, 13, 3, 5, 14, 2, 4, 15, 2, 2, 17, 1, 1, 18, 2, 2, 19, 4, 6, 20, 2, 2, 21, 4, 6, 22, 4, 6, 23, 2, 4, 24, 2, 2, 26, 1, 1, 27, 2, 2, 28, 4, 4, 29, 8, 5, 30, 2, 2, 31, 6, 8, 32

Offset: 1

Clark Kimberling, Aug 29 2013

See A228667 for the definition of accelerated continued fraction.

			The initial 2,1,1 means that both the ACF and CF of sqrt(2) have repeating strings of length 1; the next 3,2,2 means that the ACF and CF of sqrt(3) have repeating strings of length 2 and 2.  In the table below, Mathematica notation is used for repeating continued fractions; x(n) approximates sqrt(n)-ACF(sqrt(n)) and y(n) approximates sqrt(n)-CF(sqrt(n)).
n . ACF(sqrt(n)) . x(n) ........... CF(sqrt(n)) ... y(n)
2 . {1,{2}} ..... -0.32 ........... {1,{2}} ....... -0.32
3 . {2,{-4,4}} .. -1.3 x 10^(-11) . {1,{1,2}} ..... -7 x 10^(-6)
7 . {3,{-3,6}} .. -5.0 x 10^(-12) . {2,{1,1,1,4}} . -5 x 10^(-6)

Cf. A228668, A228488.

$MaxExtraPrecision = Infinity; period[seq_] := (If[Last[#1] == {} || Length[#1] == Length[seq] - 1, 0, Length[#1]] &)[NestWhileList[Rest, Rest[seq], #1 != Take[seq, Length[#1]] &, 1]]; periodicityReport[seq_] := ({Take[seq, Length[seq] - Length[#1]], period[#1], Take[#1, period[#1]]} &)[Take[seq, -Length[NestWhile[Rest[#1] &, seq, period[#1] == 0 &, 1, Length[seq]]]]]
(*output format {initial seqment,period length,period}*)
(*error messages occur if the sequence not found to be periodic.*)
aCF[rational_] := Module[{steps = {}, stop = False, i = 0, x = Numerator[rational], y = Denominator[rational], w, u, v, f, c},(*Step 1*)w = Mod[x, y]; Which[w == 0, c[i] = x/y; stop = True; AppendTo[steps, "A"], 0 < w <= y/2, c[i] = Floor[x/y]; {u, v, f} = {y, w, 1}; AppendTo[steps, "B"], w > y/2, c[i] = 1 + Floor[x/y]; {u, v, f} = {y, y - w, -1};    AppendTo[steps, "C"]];  i++; (*Step 2*)While[stop =!= True, w = Mod[u, v]; Which[f == 1 && w == 0, c[i] = u/v; stop = True; AppendTo[steps, "0.1"], f == -1 && w == 0, c[i] = -u/v; stop = True; AppendTo[steps, "0.2"], f == 1 && w <= v/2, c[i] = Floor[u/v]; {u, v, f} = {v, w, 1}; AppendTo[steps, "1"], f == 1 && w > v/2, c[i] = 1 + Floor[u/v]; {u, v, f} = {v, v - w, -1}; AppendTo[steps, "2"], f == -1 && w <= v/2, c[i] = -Floor[u/v]; {u, v, f} = {v, w, -1}; AppendTo[steps, "3"], f == -1 && w > v/2, c[i] = -1 - Floor[u/v]; {u, v, f} = {v, v - w, -f}; AppendTo[steps, "4"]]; i++]; (*Display results*) {FromContinuedFraction[#], {"Steps", steps}, {"ACF", #}, {"CF", ContinuedFraction[x/y]}} &[Map[c, Range[i] - 1]]]
m = Map[{#, Map[periodicityReport[#][[2]] &, {Drop[#[[1]][[2]], -3],    Drop[#[[2]][[2]], -3]} &[aCF[Rationalize[Sqrt[#], 10^-80]][[{3, 4}]]]]} &, Select[Range[200], ! IntegerQ[Sqrt[#]] &]]
Flatten[m] (* Peter J. C. Moses, Aug 28 2013 *)

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A228825 Delayed continued fraction of e.

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