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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.
1
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).
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
Sequence in context: A139531 A047249 A327257 * A236211 A279616 A050068
KEYWORD
nonn,cofr,more
AUTHOR
Clark Kimberling, Sep 08 2013
STATUS
approved