A015884 -id:A015884 - cp's OEIS frontend

A061233 Pierce expansion for 4 - Pi.

1, 7, 112, 115, 157, 372, 432, 1340, 7034, 8396, 9200, 18846, 29558, 34050, 89754, 101768, 1361737, 48461857, 81164005, 145676139, 163820009, 182446527, 5021656281, 8401618827, 22255558907, 28334352230, 127113921970, 310272097461, 782301280193, 5560255100022, 9925600136870, 85169484256928, 2542699818508737, 3145584963639199, 397021758001902006, 467746771316089905

Offset: 0

Frank Ellermann, May 15 2001

Also, alternating Engel expansion for Pi.
Pi = 4 - 1/1 + 1/(1*7) - 1/(1*7*112) + 1/(1*7*112*115) - ...
Pierce expansions are always strictly increasing.

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 0..1000 (terms 0 to 400 computed by T. D. Noe; terms 401 to 1000 computed by G. C. Greubel, Dec 31 2016)
Eric Weisstein's World of Mathematics, Pierce Expansion
Index entries for sequences related to Engel expansions

A014014 and A015884 are inferior versions of this sequence.

Cf. A154956 (analog for 2/Pi).

Digits := 1000: x0 := 4-Pi-4^(-1000): x1 := 4-Pi+4^(-1000): ss := []: # when expansions of x0 and x1 differ, halt
k0 := floor(1/x0): k1 := floor(1/x1): while k0=k1 do ss := [op(ss),k0]: x0 := 1-k0*x0: x1 := 1-k1*x1: k0 := floor(1/x0): k1 := floor(1/x1): od:

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[4 - Pi, 7!], 25] (* G. C. Greubel, Dec 31 2016 *)

A061233(N=199)={localprec(N); my(c=4-Pi, d=c+c/10^N, a=[1\c]); while(a[#a]==1\d&&c=1-c*a[#a], d=1-d*a[#a]; a=concat(a, 1\c)); a[^-1]} \\ optional arg fixes precision, roughly equal to total number of digits in the result. - M. F. Hasler, Nov 24 2020

More terms from Eric Rains (rains(AT)caltech.edu), May 31 2001

A228929 Optimal ascending continued fraction expansion of Pi - 3.

Original entry on oeis.org

7, -113, 4739, -46804, 134370, -614063, 1669512, -15474114, -86232481, 1080357006, -8574121305, -24144614592, 133884333083, -2239330253016, -6347915250018, 14541933941298, -42301908155404, -298013673554972, 5177473084279656, -46709468571434452, 1201667304102142095, -68508286025632748778, 850084640720511629243, -2458418086834560217354

Offset: 1

Giovanni Artico, Sep 08 2013

sign

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.

			Pi = 3 + 1/7*(1 - 1/113*(1 + 1/4739*(1 - 1/46804*(1 + 1/134370*(1 - 1/614063*(1 + 1/1669512*(1 + ...))))))).

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

Cf. A006784, A015884, A228930, A228931, A228932, A228933, A228934.

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)

# 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)

ArticoExp[x_, n_] :=  Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000}, ArticoExp[Pi, 20]]

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.

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A061233 Pierce expansion for 4 - Pi.

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