A228930 Optimal ascending continued fraction expansion of e - 2.

1, -4, 8, 67, 266, 9757, 47748, -97258, -251115, 671488, -4724169, -28356343, 125269419, -498668029, -5426804695, 15313259790, -40462770156, 105160602326, -4412226092528, -350847041434052, -54342998565206181

Offset

1,2

Comments

See A228929 for explanation.

Links

G. C. Greubel, Table of n, a(n) for n = 1..500

Formula

Given a positive real number x, let z(0) = x - floor(x) and z(k+1) = abs(z(k))*round(1/abs(z(k))) - 1; then a(n) = sign(z(n))*round(1/abs(z(n))) for n>0.

Example

e = 2 + 1*(1 - 1/4*(1 + 1/8*(1 + 1/67*(1 + 1/266*(1 + 1/9757*(1 + ...)))))).

Maple

ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
# List the first 20 terms of the expansion of exp(1)-2
ArticoExp(exp(1), 20)

Mathematica

ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000}, ArticoExp[Exp[1] - 2, 20]] (* G. C. Greubel, Dec 26 2016 *)

Crossrefs

Cf. A228929.

Sequence in context: A086891 A117636 A285634 * A051226 A013112 A274461

Adjacent sequences: A228927 A228928 A228929 * A228931 A228932 A228933

Keyword

sign,cofr

Author

Giovanni Artico, Sep 09 2013

Status

approved