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A232325 Engel expansion of 1 to the base Pi.

%I A232325 #17 Jul 18 2020 14:11:46
%S A232325 4,12,72,2111,14265,70424,308832,4371476,320218450,1101000257,
%T A232325 14020589841,102772320834,963205851651,5997003656523,50649135127796,
%U A232325 640772902021920,2101002284323870,35029677728070645,176996397541889098,1433436623499128186
%N A232325 Engel expansion of 1 to the base Pi.
%C A232325 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.
%C A232325 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.
%C A232325 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:
%C A232325 Choose values for r and b.
%C A232325 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.
%C A232325 For n = 0, 1, 2, ... define the integer a(n) = ceiling(b/f^(n)(r)) until f^n(r) = 0.
%C A232325 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.
%C A232325 For the present sequence we apply this algorithm with r := 1 and with the base b := Pi.
%C A232325 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.
%C A232325 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.
%H A232325 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A232325 Wikipedia, <a href="http://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%F A232325 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.
%F A232325 Engel series expansion of 1 to the base Pi:
%F A232325 1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
%F A232325 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.
%e A232325 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.
%e A232325 Comparison of generalized Engel expansions of 1 to the base Pi.
%e A232325 A232325: Engel series expansion of 1 to the base Pi
%e A232325 1 = Pi/4 + Pi^2/(4*12) + Pi^3/(4*12*72) + Pi^4/(4*12*72*2111) + ....
%e A232325 A232326: Pierce series expansion of 1 to the base Pi
%e A232325 1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + - ....
%e A232325 Running the algorithm with the input values r = 1 and base -Pi produces the expansion
%e A232325 1 = Pi/3 - Pi^2/(3*70) - Pi^3/(3*70*740) + Pi^4/(3*70*740*6920) + - - + ....
%e A232325 Running the algorithm with the input values r = -1 and base -Pi produces the expansion
%e A232325 1 = Pi/4 + Pi^2/(4*11) - Pi^3/(4*11*73) - Pi^4/(4*11*73*560) + + - - ....
%p A232325 # Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
%p A232325 map_iterate := proc(n,b,x) option remember;
%p A232325 if n = 0 then
%p A232325    x
%p A232325 else
%p A232325   -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
%p A232325 end if
%p A232325 end proc:
%p A232325 # Define the terms of the expansion of x to the base b
%p A232325 a := n -> ceil(evalf(b/map_iterate(n,b,x))):
%p A232325 Digits:= 500:
%p A232325 # Choose values for x and b
%p A232325 x := 1: b:= Pi:
%p A232325 seq(a(n), n = 0..19);
%Y A232325 Cf. A014014, A006784, A061233, A185565, A230601, A232326, A232327, A232328, A303877.
%K A232325 nonn,easy
%O A232325 0,1
%A A232325 _Peter Bala_, Nov 25 2013