A061233 -id:A061233 - cp's OEIS frontend

A232325 Engel expansion of 1 to the base Pi.

4, 12, 72, 2111, 14265, 70424, 308832, 4371476, 320218450, 1101000257, 14020589841, 102772320834, 963205851651, 5997003656523, 50649135127796, 640772902021920, 2101002284323870, 35029677728070645, 176996397541889098, 1433436623499128186

Offset: 0

Peter Bala, Nov 25 2013

Let r and b be positive real numbers. We define an Engel expansion of r to the base b to be a (possibly infinite) nondecreasing sequence of positive integers [a(0), a(1), a(2), ...] such that we have the series representation r = b/a(0) + b^2/(a(0)*a(1)) + b^3/(a(0)*a(1)*a(2)) + .... Depending on the values of r and b such an expansion may not exist, and if it does exist it may not be unique.
When b = 1 we recover the ordinary Engel expansion of r. See A181565 and A230601 for some predictable Engel expansions to a base b other than 1.
In the particular case that the base b >= 1 and 0 < r < b then we can find an Engel expansion of r to the base b using the following algorithm:
Choose values for r and b.
Define the map f(x) (which depends on the base b) by f(x) = x/b*ceiling(b/x) - 1 and let f^(n)(x) denote the n-th iterate of the map f(x), with the convention that f^(0)(x) = x.
For n = 0, 1, 2, ... define the integer a(n) = ceiling(b/f^(n)(r)) until f^n(r) = 0.
When b >= 1 and 0 < r < b the sequence a(n) produced by this algorithm provides an Engel expansion of r to the base b.
For the present sequence we apply this algorithm with r := 1 and with the base b := Pi.
We can also get an alternating series representation for r in powers of b (still assuming b >= 1 and 0 < r < b), called a Pierce series expansion of r to the base b, by running the above algorithm but now with input values -r and base b. See A232326.
In addition, we can obtain two further series expansions for r in powers of b by running the algorithm with either the input values r and base -b or with the input values -r and base -b. See examples below. See A232327 and A232328 for other examples of these types of expansions.

			Truncation F_5(z) = 1 - ( z/4 + z^2/(4*12) + z^3/(4*12*72) + z^4/(4*12*72*2111) + z^5/(4*12*72*2111*14265) ). The polynomial has a positive real zero at z = 3.14159 26535 (9...), which agrees with Pi to 10 decimal places.
Comparison of generalized Engel expansions of 1 to the base Pi.
A232325: Engel series expansion of 1 to the base Pi
1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
A232326: Pierce series expansion of 1 to the base Pi
1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + - ....
Running the algorithm with the input values r = 1 and base -Pi produces the expansion
1 = Pi/3 - Pi^2/(3*70) - Pi^3/(3*70*740) + Pi^4/(3*70*740*6920) + - - + ....
Running the algorithm with the input values r = -1 and base -Pi produces the expansion
1 = Pi/4 + Pi^2/(4*11) - Pi^3/(4*11*73) - Pi^4/(4*11*73*560) + + - - ....

Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion

Cf. A014014, A006784, A061233, A185565, A230601, A232326, A232327, A232328, A303877.

# Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
map_iterate := proc(n,b,x) option remember;
if n = 0 then
   x
else
  -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
end if
end proc:
# Define the terms of the expansion of x to the base b
a := n -> ceil(evalf(b/map_iterate(n,b,x))):
Digits:= 500:
# Choose values for x and b
x := 1: b:= Pi:
seq(a(n), n = 0..19);

a(n) = ceiling(Pi/f^(n)(1)), where f^(n)(x) denotes the n-th iterate of the map f(x) = x/Pi*(ceiling(Pi/x)) - 1, with the convention that f^(0)(x) = x.
Engel series expansion of 1 to the base Pi:
1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
The associated power series F(z) := 1 - ( z/4 + z^2/(4*12) + z^3/(4*12*72) + z^4/(4*12*72*2111) + ...) has a zero at z = Pi. Truncating the series F(z) to n terms produces a polynomial F_n(z) with rational coefficients which has a real zero close to Pi. See below for an example.

A154956 Pierce expansion of 2/Pi.

Original entry on oeis.org

1, 2, 3, 5, 10, 71, 868, 1788, 7455, 44266, 54626, 74153, 224166, 390471, 1489304, 3737961, 22277163, 37201631, 113275744, 165029426, 2642368758, 3362202939, 5191046363, 8438525012, 36226438506, 40174126779, 125336047846, 531802867080, 599020778171

Offset: 0

Jaume Oliver Lafont, Jan 18 2009

nonn

			1 - 1/2(1 - 1/3(1 - 1/5(1 - 1/10(1 - 1/71)))) = 2/(355/113).

Simon Plouffe and G. C. Greubel, Table of n, a(n) for n = 0..500 (terms from 0 to 216 were computed by Simon Plouffe)

Cf. A006283 (1/Pi), A061233 (4 - Pi).

Cf. A060294 (decimal expansion of 2/Pi). - R. J. Mathar, Jan 21 2009

Digits := 300: Pierce := proc(x) local resid,a,i,an ; resid := x ; a := [] ; for i from 1 do an := floor(1./resid) ; a := [op(a),an] ; resid := evalf(1.-an*resid) ; if ilog10( mul(i,i=a)) > 0.7*Digits then break ; fi ; od: RETURN(a) ; end: a060294 := evalf(2/Pi) ; Pierce(a060294) ; # R. J. Mathar, Jan 21 2009

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[2/Pi, 7!], 50] (* G. C. Greubel, Nov 13 2016 *)

A154956(N=99)={localprec(N); my(c=2/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]}  \\ The optional argument is the precision used, approx. equal to the total number of digits in the result. - M. F. Hasler, Jul 04 2016

More terms from R. J. Mathar, Jan 21 2009

Offset in b-file corrected by N. J. A. Sloane, Aug 31 2009

A014014 Alternating Engel expansion of Pi.

Original entry on oeis.org

3, 7, 112, 115, 157, 372, 432, 1340, 7034, 8396, 9200, 18846, 29558, 34050, 89754, 101768, 1361737, 48461857, 81164005, 145676139, 163820009, 182446527, 5021656281, 8401618827, 22255558907

Offset: 0

Simon Plouffe

nonn

Index entries for sequences related to Engel expansions

A061233 is a better version of this sequence.

A015884 A modified Pierce-type expansion for Pi: Pi = a(0) + Sum_{n>=1} (-1)^floor(n/2)/(Product_{i=1..n} a(i)).

Original entry on oeis.org

3, 7, 113, 4739, 46804, 134370, 614063, 1669512, 15474115, 18858140, 19180902, 41486462, 492988666, 1794101482, 34644610027, 48670872793, 97414216753, 138669015304, 195575194804, 543142431219, 3173502039447, 4968328076747

Offset: 0

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), Jun 02 2000

nonn

			Pi = 3 + 1/7 - 1/(7*113) - 1/(7*113*4739) + 1/(7*113*4739*46804) + 1/(7*113*4739*46804*134370) - 1/(7*113*4739*46804*134370*614063) - 1/(7*113*4739*46804*134370*614063*1669512) + ...
From _M. F. Hasler_, Apr 09 2018: (Start)
Using the formulas given in the formula section, we get:
a(0) = 3, A(1) = Pi-3 = 0.14159..., a(1) = floor(1/A(1)) = floor(7.0626...) = 7,
A(2) = 1 - A(1)*a(1) = 0.00885..., a(2) = ceiling(1/A(2)) = 113,
A(3) = A(2)*a(2) - 1 = 0.000221..., a(3) = floor(1/A(3)) = 4739,
A(4) = 1 - A(3)*a(3) = 2.136585...e-5, a(4) = ceiling(1/A(2)) = 46804,
A(5) = A(4)*a(4) - 1 = 7.442125...e-7, a(5) = floor(1/A(3)) = 134370, ... (End)

Index entries for sequences related to Engel expansions

Cf. A061233.

{A=Pi-a=3; for(n=0,oo, print1(a","); A=abs(1-A*a=if(bittest(n,0),ceil(1/A),1\A)))} \\ M. F. Hasler, Apr 09 2018

a(0) = floor(Pi); A(1) = Pi-a(0); a(2*n-1) = floor(1/A(2*n-1)); A(2*n) = 1-a(2*n-1)*A(2*n-1); a(2*n) = ceiling(1/A(2*n)) and A(2*n+1) = a(2*n)*A(2*n)-1 for n >= 1.

Better description and more terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 28 2001

A140076 Pierce expansion of the cube root of 1/2.

Original entry on oeis.org

1, 4, 5, 7, 8, 18, 384, 7958, 14304, 16623, 18610, 20685, 72923, 883177, 1516692, 2493788, 2504069, 22881179, 110219466, 2241255405, 34982468090, 64356019489, 110512265214, 1142808349967, 3550630472116, 5238523454726, 7129035664265

Offset: 1

Gerard P. Michon, Jun 01 2008

2^(-1/3) = 1-1/4(1-1/5(1-1/7(1-1/8(1-1/18(1-1/384(...))))))

			a(1) is 1 because the floor of 2^(1/3) is 1.
a(2)=4 because 1/(1-2^(-1/3)) is 4.8473221...

G. C. Greubel, Table of n, a(n) for n = 1..500
G. P. Michon, Pierce Expansions.
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A091831, A006283, A006284, A061233, A118242.

$MaxExtraPrecision = 80; x[1] = 2^(-1/3); a[n_] := a[n] = Floor[1/x[n]]; x[n_] := x[n] = 1 - a[n-1]*x[n-1]; Table[a[n], {n, 1, 27}] (* Jean-François Alcover, Dec 12 2011 *)
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[2^(-1/3), 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

Starting with x(1)=2^(-1/3), a(n) = floor(1/x(n)) and x(n+1) = 1-a(n)x(n).

A232326 Pierce expansion of 1 to the base Pi.

Original entry on oeis.org

3, 69, 310, 1017, 36745, 214369, 966652, 11159821, 74039764, 550021544, 4481549430, 16543857917, 87205978613, 476981856953, 30989048525367, 203786458494160, 711639924282497, 3174772986229899, 29814569078896025, 100158574806804154

Offset: 0

Peter Bala, Nov 26 2013

Let r and b be positive real numbers. We define a Pierce expansion of r to the base b to be a (possibly infinite) increasing sequence of positive integers [a(0), a(1), a(2), ...] such that we have the alternating series representation r = b/a(0) - b^2/(a(0)*a(1)) + b^3/(a(0)*a(1)*a(2)) - .... Depending on the values of r and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 and 0 < r < 1 we recover the ordinary Pierce expansion of r.
See A058635, A192223 and A230600 for some predictable Pierce expansions to a base b other than 1.
In the particular case that the base b >= 1 and 0 < r < b then we can find a Pierce expansion of r to the base b as follows:
Define the map f(x) (which depends on the base b) by f(x) = x/b*ceiling(b/x) - 1 and let f^(n)(x) denote the n-th iterate of the map f(x), with the convention that f^(0)(x) = x.
For n = 0,1,2,... define a(n) = ceiling(b/f^(n)(-r)) until f^n(-r) = 0.
Then it can be shown that the sequence of positive integers |a(n)| is a Pierce expansion of r to the base b.
For the present sequence we apply this algorithm with r := 1 and with base b := Pi. See A232325 for an Engel expansion of 1 to the base Pi.

Eric Weisstein's World of Mathematics, Pierce Expansion

Cf. A014014, A006784, A058635, A061233, A192223, A230600, A232325, A232327, A232328.

# Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
map_iterate := proc(n,b,x) option remember;
if n = 0 then
   x
else
   -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
end if
end proc:
# Define the (signed) terms of the expansion of x to the base b
a := n -> ceil(evalf(b/map_iterate(n,b,x))):
Digits:= 500:
# Choose values for x and b
x := -1: b:= Pi:
seq(abs(a(n)), n = 0..19);

a(n) = ceiling(Pi/f^(n)(-1)), where f^(n)(x) denotes the n-th iterate of the map f(x) = x/Pi*ceiling(Pi/x) - 1, with the convention that f^(0)(x) = x.
Pierce series expansion of 1 to the base Pi:
1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + ....
The associated power series F(z) := 1 - ( z/3 - z^2/(3*69) + z^3/(3*69*310) - z^4/(3*69*310*1017) + ...) has a zero at z = Pi. Truncating the series F(z) to n terms produces a polynomial F_n(z) with rational coefficients which has a real zero close to Pi.

cp's OEIS Frontend

A232325 Engel expansion of 1 to the base Pi.

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A154956 Pierce expansion of 2/Pi.

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A014014 Alternating Engel expansion of Pi.

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A015884 A modified Pierce-type expansion for Pi: Pi = a(0) + Sum_{n>=1} (-1)^floor(n/2)/(Product_{i=1..n} a(i)).

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A140076 Pierce expansion of the cube root of 1/2.

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A232326 Pierce expansion of 1 to the base Pi.

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