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Optimal ascending continued fraction expansion of e - 2.
2

%I #13 Dec 27 2016 02:35:19

%S 1,-4,8,67,266,9757,47748,-97258,-251115,671488,-4724169,-28356343,

%T 125269419,-498668029,-5426804695,15313259790,-40462770156,

%U 105160602326,-4412226092528,-350847041434052,-54342998565206181

%N Optimal ascending continued fraction expansion of e - 2.

%C See A228929 for explanation.

%H G. C. Greubel, <a href="/A228930/b228930.txt">Table of n, a(n) for n = 1..500</a>

%F 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.

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

%p 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

%p # List the first 20 terms of the expansion of exp(1)-2

%p ArticoExp(exp(1),20)

%t 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 *)

%Y Cf. A228929.

%K sign,cofr

%O 1,2

%A _Giovanni Artico_, Sep 09 2013