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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.

The Engel expansion of 1/Pi is given in A014012 and the Pierce (or alternating Engel) expansion of 1/Pi is found in A006283.

We can unify the algorithms for finding the Engel and Pierce expansions of a real number as follows.

Define the map f:[-1,1]\{0} -> (-1/2,1) by f(x) = x*ceiling(1/x) - 1 and let f^(n)(x) denote the n-th iterate of f, with the convention that f^(0)(x) = x. Let r be a real number such that 0 < r < 1.

Then the sequence of positive integers e(n) := ceiling(1/f^(n)(r)) is the Engel expansion of r. The associated Engel series representation is r = 1/e(0) + 1/(e(0)*e(1)) + 1/(e(0)*e(1)*e(2)) + ....

The sequence of positive integers p(n) := |ceiling(1/f^(n)(-r))| is the Pierce expansion of r. The associated Pierce series representation is r = 1/p(0) - 1/(p(0)*p(1)) + 1/(p(0)*p(1)*p(2)) - ....

By working with a modification of the map f we can find two generalized Engel-type expansions for the real number r (still assuming 0 < r < 1). To this end, we define the map g:[-1,1]\{0} -> (-1/2,1) 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.

A)

Our first generalized expansion of r is the integer sequence a(n) := |ceiling(1/g^(n)(r))| for n >= 0 and until g^n(r) = 0. It can be shown that we have a generalized Engel-type representation for r by means of the (possibly infinite) series r = 1/a(0) - 1/(a(0)*a(1)) - 1/(a(0)*a(1)*a(2)) + 1/(a(0)*a(1)*a(2)*a(3)) + 1/(a(0)*a(1)*a(2)*a(3)*a(4)) - - + + ..., where the pattern of signs of the terms is as indicated.

The series will be finite if and only if r is rational.

The present sequence is an example of this first type of generalized Engel expansion for the real number r := 1/Pi.

B)

The second generalized Engel expansion of r is defined as the sequence of integers b(n) := |ceiling(1/g^(n)(-r))| for n >= 0 and until g^(n)(-r) = 0.

It can be shown that we now have a generalized Engel-type representation for r of the form r = 1/b(0) + 1/(b(0)*b(1)) - 1/(b(0)*b(1)*b(2)) - 1/(b(0)*b(1)*b(2)*b(3)) + + - - ....

Again, the series terminates when r is rational, otherwise it is infinite.

See A232328 for the generalized Engel expansion of 1/Pi of the second kind.

We can generalize the Engel and Pierce expansions of a real number even further by considering series expansions to a base b. See A232325 for a definition and details. The usual Engel and Pierce expansions occur when the base b = 1 and the two generalized Engel expansions described above arise when the base b = -1.

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.