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 A228929 Optimal ascending continued fraction expansion of Pi - 3. 6
 7, -113, 4739, -46804, 134370, -614063, 1669512, -15474114, -86232481, 1080357006, -8574121305, -24144614592, 133884333083, -2239330253016, -6347915250018, 14541933941298, -42301908155404, -298013673554972, 5177473084279656, -46709468571434452, 1201667304102142095, -68508286025632748778, 850084640720511629243, -2458418086834560217354 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Definition of the expansion: for a positive real number x, there is always a unique sequence of signed integers with increasing absolute value |a(i)|>|a(i-1)| such that x =floor(x)+ 1/a(1) + 1/a(1)/a(2) + 1/a(1)/a(2)/a(3) + 1/a(1)/a(2)/a(3)/a(4) ... or equivalently x=floor(x)+1/a(1)*(1+1/a(2)*(1+1/a(3)*(1+1/a(4)*(1+...)))) giving the fastest converging series with this representation. This formula can be represented as a regular ascending continued fraction. The expansion is similar to Engel and Pierce expansions, but the sign of the terms is not predefined and determined by the algorithm for optimizing the convergence. For x rational number the expansion has a finite number of terms, for x irrational an infinite number. Empirically the sequence doesn't show any evident regularity except in some interesting cases. 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 Pi = 3 + 1/7*(1 - 1/113*(1 + 1/4739*(1 - 1/46804*(1 + 1/134370*(1 - 1/614063*(1 + 1/1669512*(1 + ...))))))). MAPLE # Slow procedure valid for every number ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := n-floor(n); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(evalf(z))]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc # Fast procedure, not suited for rational numbers 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 Pi ArticoExp(Pi, 20) MATHEMATICA ArticoExp[x_, n_] :=  Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{\$MaxExtraPrecision = 50000}, ArticoExp[Pi, 20]] PROG (DERIVE) ArticoExp(x, n) := VECTOR(ROUND(1, ABS(k))*SIGN(k), k, ITERATES(ROUND(1, ABS(u))*ABS(u) - 1, u, MOD(x), n)) Precision:=Mixed PrecisionDigits:=10000 ArticoExp(PI, 20) CROSSREFS Cf. A006784, A015884, A228930, A228931, A228932, A228933, A228934. Sequence in context: A152927 A064330 A159552 * A086788 A199672 A240288 Adjacent sequences:  A228926 A228927 A228928 * A228930 A228931 A228932 KEYWORD sign AUTHOR Giovanni Artico, Sep 08 2013 STATUS approved

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Last modified January 28 22:40 EST 2020. Contains 331328 sequences. (Running on oeis4.)