A303877 -id:A303877 - 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.

A283735 Expansion of 10 in base Pi.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 0, 1, 2, 0, 1, 0, 2, 2, 1, 2, 0, 2, 1, 1, 0, 0, 1, 1, 1, 2, 1, 2, 0, 0, 3, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 2, 1, 0, 0, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 3, 0, 0, 2, 0, 1, 0, 1

Offset: 3

José Eduardo Gaboardi de Carvalho, Mar 15 2017

			100.0102212222..._{Pi}

José Eduardo Gaboardi de Carvalho, Java program

CF. A000796, A303877.

RealDigits[10, Pi, 100][[1]] (* Indranil Ghosh, Mar 15 2017 *)

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A232325 Engel expansion of 1 to the base Pi.

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