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A232326 Pierce expansion of 1 to the base Pi.

%I A232326 #12 Feb 16 2025 08:33:20
%S A232326 3,69,310,1017,36745,214369,966652,11159821,74039764,550021544,
%T A232326 4481549430,16543857917,87205978613,476981856953,30989048525367,
%U A232326 203786458494160,711639924282497,3174772986229899,29814569078896025,100158574806804154
%N A232326 Pierce expansion of 1 to the base Pi.
%C A232326 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.
%C A232326 See A058635, A192223 and A230600 for some predictable Pierce expansions to a base b other than 1.
%C A232326 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:
%C A232326 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 A232326 For n = 0,1,2,... define a(n) = ceiling(b/f^(n)(-r)) until f^n(-r) = 0.
%C A232326 Then it can be shown that the sequence of positive integers |a(n)| is a Pierce expansion of r to the base b.
%C A232326 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.
%H A232326 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>
%F A232326 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 A232326 Pierce series expansion of 1 to the base Pi:
%F A232326 1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + ....
%F A232326 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.
%p A232326 # Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
%p A232326 map_iterate := proc(n,b,x) option remember;
%p A232326 if n = 0 then
%p A232326    x
%p A232326 else
%p A232326    -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
%p A232326 end if
%p A232326 end proc:
%p A232326 # Define the (signed) terms of the expansion of x to the base b
%p A232326 a := n -> ceil(evalf(b/map_iterate(n,b,x))):
%p A232326 Digits:= 500:
%p A232326 # Choose values for x and b
%p A232326 x := -1: b:= Pi:
%p A232326 seq(abs(a(n)), n = 0..19);
%Y A232326 Cf. A014014, A006784, A058635, A061233, A192223, A230600, A232325, A232327, A232328.
%K A232326 nonn,easy
%O A232326 0,1
%A A232326 _Peter Bala_, Nov 26 2013