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

Author

Clark Kimberling, Sep 08 2013

Keywords

• nonn
• cofr
• more

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).

Examples

			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).
		

Crossrefs

Cf. A228825, A014209.

Programs

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}]]

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