cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A232325 Engel expansion of 1 to the base Pi.

Original entry on oeis.org

4, 12, 72, 2111, 14265, 70424, 308832, 4371476, 320218450, 1101000257, 14020589841, 102772320834, 963205851651, 5997003656523, 50649135127796, 640772902021920, 2101002284323870, 35029677728070645, 176996397541889098, 1433436623499128186
Offset: 0

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Author

Peter Bala, Nov 25 2013

Keywords

Comments

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.

Examples

			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) + + - - ....

Links

Crossrefs

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

Programs

  • Maple
    # 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);

Formula

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.

A061233 Pierce expansion for 4 - Pi.

Original entry on oeis.org

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

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Author

Frank Ellermann, May 15 2001

Keywords

Comments

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.

Links

Crossrefs

A014014 and A015884 are inferior versions of this sequence.
Cf. A154956 (analog for 2/Pi).

Programs

  • Maple
    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:
  • Mathematica
    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 *)
  • PARI
    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

Extensions

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

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

Views

Author

Peter Bala, Nov 26 2013

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Maple
    # 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);

Formula

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.
Showing 1-3 of 3 results.

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