A133570 -id:A133570 - 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)

A228825 Delayed continued fraction of e.

Original entry on oeis.org

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}]

A228936 Expansion of (1 + 3x - 3x^3 - x^4)/(1 + 2*x^2 + x^4).

Original entry on oeis.org

1, 3, -2, -9, 2, 15, -2, -21, 2, 27, -2, -33, 2, 39, -2, -45, 2, 51, -2, -57, 2, 63, -2, -69, 2, 75, -2, -81, 2, 87, -2, -93, 2, 99, -2, -105, 2, 111, -2, -117, 2, 123, -2, -129, 2, 135, -2, -141, 2, 147

Offset: 0

Giovanni Artico, Oct 26 2013

Optimal simple continued fraction (with signed denominators) of exp(1/3). See A228935.
The convergents are a subset of those of the standard regular continued fraction; the sequence of the signs of the difference between the convergents and exp(1/3) starts with: -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, ...
For every couple of successive equal signs in this sequence there is a convergent of the standard expansion not present in this one.
Repeating the expansion for other numbers of type 1/k a common pattern seems to emerge. Examples:
exp(1/4) gives 1, 4, -2, -12, 2, 20, -2, -28, 2, 36, -2, -44, 2, 52, ...
exp(1/5) gives 1, 5, -2, -15, 2, 25, -2, -35, 2, 45, -2, -55, 2, 65, ...
so it seems that in general the terms for exp(1/k) are generated by the formulas a(0)=1, a(2n+1) = (-1)^n*k*(2n+1) for n >= 0, a(2n) = (-1)^n*2 for n > 0. These formulas give this expansion for exp(1/k):
exp(1/k) = 1+1/(k+1/(-2+1/(-3k+1/(2+1/(5k+1/(-2+1/(-7k+1/(2+...)))))))).
which can be rewritten in this equivalent form:
exp(1/k) = 1+1/(k-1/(2+1/(3k-1/(2+1/(5k-1/(2+1/(7k-1/(2+...)))))))).
This general expansion seems to be valid for any real value of k.
Closed form for the general case exp(1/k): b(n) = (1+(-1)^n-(1-(-1)^n)*k*n/2)*i^(n*(n+1)) for n>0 and with i=sqrt(-1). [Bruno Berselli, Nov 01 2013]

			exp(1/3) = 1+1/(3+1/(-2+1/(-9+1/(2+1/(15+1/(-2+1/(-21+1/(2+...)))))))) or
exp(1/3) = 1+1/(3-1/(2+1/(9-1/(2+1/(15-1/(2+1/(21-1/(2+...))))))))

Index entries for linear recurrences with constant coefficients, signature (0,-2,0,-1).

Cf. A133593, A133570, A228935.

SCF := proc (n, q::posint)::list; local L, i, z; Digits := 10000; L := [round(n)]; z := n; for i from 2 to q do if z = op(-1, L) then break end if; z := 1/(z-op(-1, L)); L := [op(L), round(z)] end do; return L end proc
SCF(exp(1/3), 50)  # Giovanni Artico, Oct 26 2013

Vec(-(x-1)*(x+1)*(x^2+3*x+1)/(x^2+1)^2+O(x^100)) \\ Colin Barker, Oct 26 2013

This sequence can be generated by these formulas:
a(0)=1; for n >= 0, a(2n+1) = 3*(-1)^n*(2n+1), a(2n) = 2*(-1)^n for n > 0.
Formulae for the general case exp(1/k):
b(0)=1; for n >= 0, b(2n+1) = (-1)^n*k*(2n+1), b(2n) = 2*(-1)^n.
b(n) = 2*cos(n*Pi/2) + k*n*sin(n*Pi/2) for n > 0.
exp(1/k) = 1+1/(k-1/(2+1/(3k-1/(2+1/(5k-1/(2+1/(7k-1/(2+...)))))))).
G.f.: (1-x)*(1+x)*(1+k*x+x^2)/(1+x^2)^2.
From Colin Barker, Oct 26 2013: (Start)
a(n) = (-i)^n + i^n + (1/2)*(((-i)^n-i^n)*n)*(3*i) for n > 0, where i=sqrt(-1).
a(2n) = 2*(3*n*sin(Pi*n) + cos(Pi*n)) for n > 0.
a(2n+1) = (6*n+3)*cos(Pi*n) - 2*sin(Pi*n) for n >= 0.
a(n) = -2*a(n-2) - a(n-4) for n > 4.
G.f.: -(x-1)*(x+1)*(x^2+3*x+1) / (x^2+1)^2. (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

A228937 Expansion of (1+2x+30x^2+13x^3-13x^5-30x^6-2x^7-x^8)/(1+2*x^4+x^8).

Original entry on oeis.org

1, 2, 30, 13, -2, -17, -90, -28, 2, 32, 150, 43, -2, -47, -210, -58, 2, 62, 270, 73, -2, -77, -330, -88, 2, 92, 390, 103, -2, -107, -450, -118, 2, 122, 510, 133, -2, -137, -570, -148, 2, 152, 630, 163, -2, -167, -690, -178, 2, 182

Offset: 0

Giovanni Artico, Oct 28 2013

Optimal simple continued fraction (with signed denominators) of exp(2/5)
See A228935.
The convergents are a subset of those of the standard regular continued fraction; the sequence of the signs of the difference between the convergents and exp(2/5) starts with:
-1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1,...
For every couple of successive equal signs in this sequence there is a convergent of the standard expansion not present in this one.
Repeating the expansion for other numbers of type 2/k a common pattern seems to emerge. Examples:
exp(2/7) gives 1, 3, 42, 18, -2, -24, -126, -39, 2, 45, 210, 60, -2,...
exp(2/9) gives 1, 4, 54, 23, -2, -31, -162, -50, 2, 58, 270, 77, -2,...
so it seems that in general the terms for exp(2/k) are generated by the following formulas:
b(0)=1, b(1)=k/2-1/2, b(2)=6*k, b(3)=5*k/2+1/2, b(4)=-2, b(5)=7*k/2+1/2, b(6)=-18*k, b(7)=-11*k/2-1/2, b(8)=2; b(n) = -2*b(n-4) -b(n-8) for n>8, recurrence which corresponds to the g.f. 1/2*(1-x)*(1+x)*(2*(1+x^6)+(k-1)*(x+x^5)+(12*k+2)*(x^2+x^4)+6*k*x^3)/(1+x^4)^2; also:
b(0)=1 , b(4m+1)=(-1)^m*((k-1)/2+3*k*m), b(4m+3)=(-1)^m*((5*k+1)/2+3*k*m), b(4m+2)=(-1)^m*(6*k+12*k*m), b(4m+4)=(-1)^(m+1)*2 for n>=0.
These formulas give this expansion for exp(2/k):
exp(2/k)=1+1/((k-1)/2+1/(6k+1/((5k+1)/2+1/(-2+1/(-(7k-1)/2+1/...)))))
that can be rewritten in this equivalent form:
exp(2/k)=1+1/(k/2-1/2+1/(6k+1/(5k/2+1/2-1/(2+1/(7x/2-1/2+1/...))))).
This general expansion seems to be valid for any real value of k.

			exp(2/5)=1+1/(2+1/(30+1/(13+1/(-2+1/(-17+1/(-90+1/(-28+1/(2+...)))))))),
or equivalently:
exp(2/5)=1+1/(2+1/(30+1/(13-1/(2+1/(17+1/(90+1/(28-1/(2+...)))))))).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-2,0,0,0,-1).

Cf. A133593, A133570, A228935, A228936.

SCF := proc (n, q::posint)::list; local L, i, z; Digits := 10000; L := [round(n)]; z := n; for i from 2 to q do if z = op(-1, L) then break end if; z := 1/(z-op(-1, L)); L := [op(L), round(z)] end do; return L end proc
SCF(exp(2/5), 50)

Join[{1}, LinearRecurrence[{0, 0, 0, -2, 0, 0, 0, -1}, {2, 30, 13, -2, -17, -90, -28, 2}, 50]] (* Bruno Berselli, Nov 06 2013 *)

G.f.: (1+2*x+30*x^2+13*x^3-13*x^5-30*x^6-2*x^7-x^8)/(1+2*x^4+x^8).
a(0)=1, a(1)=2, a(2)=30, a(3)=13, a(4)=-2, a(5)=-17, a(6)=-90, a(7)=-28, a(8)=2; for n>8, a(n) = -2*a(n-4) -a(n-8).
a(0)=1 , a(4m+1) = (-1)^m*(2+15*m), a(4m+3) = (-1)^m*(13+15*m), a(4m+2) = (-1)^m*(30+60*m), a(4m+4) = 2*(-1)^(m+1) for m>=0.

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A228826 Delayed continued fraction of sqrt(2).

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A228825 Delayed continued fraction of e.

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A228936 Expansion of (1 + 3*x - 3*x^3 - x^4)/(1 + 2*x^2 + x^4).

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

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A228937 Expansion of (1+2*x+30*x^2+13*x^3-13*x^5-30*x^6-2*x^7-x^8)/(1+2*x^4+x^8).

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A228936 Expansion of (1 + 3x - 3x^3 - x^4)/(1 + 2*x^2 + x^4).

A228937 Expansion of (1+2x+30x^2+13x^3-13x^5-30x^6-2x^7-x^8)/(1+2*x^4+x^8).