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.

A232328 A generalized Engel expansion of 1/Pi.

Original entry on oeis.org

4, 3, 6, 12, 51, 146, 280, 482, 687, 3825, 5646, 30904, 120121, 1344923, 2340376, 4456271, 194324055, 219784933, 976224357, 11584437417, 26402463827, 34635051144, 85031207055, 95014277980, 257962314442
Offset: 0

Views

Author

Peter Bala, Nov 27 2013

Keywords

Comments

For a description of two kinds of generalized Engel expansion of a real number see A232327. Compare with A006283 and A014012.

Links

Crossrefs

Cf. A006283, A014012, A232327.

Programs

  • Maple
    #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 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/Pi: b:= -1:
seq(abs(a(n)), n = 0..24);

Formula

Define the map g(x) by g(x) = -x*ceiling(-1/x) - 1 and let g^n(x) denote the n-th iterate of g, with the convention that g^0(x) = x. Then a(n) = |ceiling(1/g^n(-1/Pi))| for n >= 0.
Generalized Engel series expansion: 1/Pi = 1/4 + 1/(4*3) - 1/(4*3*6) - 1/(4*3*6*12) + 1/(4*3*6*12*51) + 1/(4*3*6*12*51*146) - - + +.

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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