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

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

1,2

Comments

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).

Links

Table of n, a(n) for n=1..28.

Formula

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

Example

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).

Mathematica

$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}]]

Crossrefs

Cf. A228825, A014209.

Sequence in context: A139531 A047249 A327257 * A236211 A279616 A050068

Adjacent sequences: A228938 A228939 A228940 * A228942 A228943 A228944

Keyword

nonn,cofr,more

Author

Clark Kimberling, Sep 08 2013

Status

approved