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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.

Product_{n>0} (1 + 1/a(n)) = 3 - phi = A094874, where phi = (1+sqrt(5))/2 is the golden mean.

From Peter Bala, Oct 28 2013: (Start)

Compare with A230600(n) = Lucas(2^n - 1).

Let x and b be positive real numbers. We define a Pierce expansion of x to the base b to be a (possibly infinite) increasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the alternating series representation x = b/a(1) - b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) - .... This definition generalizes the ordinary Pierce expansion of a real number 0 < x < 1, where the base b has the value 1. Depending on the values of x and b such a generalized Pierce expansion to the base b may not exist, and if it does exist it may not be unique.

Let Phi := 1/2*(sqrt(5) - 1) denote the reciprocal of the golden ratio. This sequence, apart from the initial term, provides a Pierce expansion of Phi^4 to the base Phi. That is we have the identity Phi^4 = Phi/4 - Phi^2/(4*11) + Phi^3/(4*11*76) - Phi^4/(4*11*76*3571) + ....

This result can be extended in two ways. Firstly, for k odd, the sequence {Lucas(k*(2^n + 1))}n>=1 gives a Pierce expansion of Phi^(4*k) to the base Phi^k. Secondly, for n = 1,2,3,..., the sequence [a(n),a(n+1),a(n+2),...] gives a Pierce expansion of Phi^(2^n + 2) to the base Phi. See below for some examples. (End)

Compare with A192223(n) = Lucas(2^n + 1).

Let x and b be positive real numbers. We define a Pierce expansion of x to the base b to be a (possibly infinite) increasing sequence of positive integers [a(1), a(2), a(3), ...] such that we have the alternating series representation x = b/a(1) - b^2/(a(1)*a(2)) + b^3/(a(1)*a(2)*a(3)) - .... This definition generalizes the ordinary Pierce expansion of a real number 0 < x < 1, where the base b is taken equal to 1. Depending on the values of x and b such a generalized Pierce expansion to the base b may not exist, and if it does exist it may not be unique.

Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence, apart from the initial term, provides a Pierce expansion of 1 to the base phi. That is, we have the identity 1 = phi/1 - phi^2/(1*4) + phi^3/(1*4*29) - phi^4/(1*4*29*1364) + ....

This result can be extended in two ways. Firstly, for k odd, the sequence {Lucas(k*(2^n - 1))} n>=1 gives a Pierce expansion of 1 to the base phi^k. Secondly, for n = 1,2,3,..., the sequence [a(n),a(n+1),a(n+2),...] provides a Pierce expansion of the quadratic irrational 1/phi^(2^n - 2) = Fibonacci(2^n - 1) - Fibonacci(2^n - 2)*phi to the base phi. Some examples are given below.

Let phi := 1/2*(1 + sqrt(5)) denote the golden ratio. This sequence, apart from the initial term, gives an Engel expansion of 1 to the base phi^2 (see A230601 for a definition of this term). The associated Engel series expansion of 1 to the base phi^2 begins 1 = phi^2/3 + phi^4/(3*18) + phi^6/(3*18*843) + phi^8/(3*18*843*1860498) + .... This result can be extended in two ways. Firstly, the sequence Lucas(2*k*(2^n - 1)) for k = 1,2,3,... is an Engel expansion of 1 to the base phi^(2*k). Secondly, for n = 1,2,3,... the sequence [a(n),a(n+1),a(n+2),...] is an Engel expansion of phi^(4 - 2^n) to the base phi^2. Some example are given below.