OFFSET
0,1
COMMENTS
From Peter Bala, Nov 22 2012: (Start)
For x in the open interval (0,1) define the map f(x) = 1 - x*floor(1/x). The n-th term (n >= 0) in the Pierce expansion of x is given by floor(1/f^(n)(x)), where f^(n)(x) denotes the n-th iterate of the map f, with the convention that f^(0)(x) = x.
Let x = (sqrt(5) - 1)/2, the reciprocal of the golden ratio, and let X = (3 - sqrt(5))/2 so that X = x^2. The Pierce expansion of X^(3^n) is [a(2*n), a(2*n+1), a(2*n+2), ...]. The Pierce expansion of x is A118242 = [1, a(0), a(1), a(2), ...]. The Pierce expansion of x^3 is [a(1), a(2), a(3), ...]. In general, the Pierce expansion of x^(3^n) for n >= 1 is [a(1)*a(3)*...*a(2*n-1), a(2*n), a(2*n+1), a(2*n+2), ...] = [sqrt(a(2*n) - 1), a(2*n), a(2*n+1), a(2*n+2), ...]. Some examples of the associated alternating series are given below.
(End)
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..15
T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly, Vol. 36 No. 10, (1929) p.523-525.
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Eric Weisstein's World of Mathematics, Pierce Expansion
FORMULA
Let c(0)=3, c(n+1) = c(n)^3-3*c(n) [A001999]; then this sequence is c(0)-1, c(0)+1, c(1)-1, c(1)+1, c(2)-1, c(2)+1, ......
a(n) = 2*F(2*3^floor(n/2)+1)-F(2*3^floor(n/2))-(-1)^n where F(k) denotes the k-th Fibonacci number A000045(k)
Let u(0)=(1+sqrt(5))/2 and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)). - Benoit Cloitre, Mar 09 2004
a(2*n) = ((3 + sqrt(5))/2)^(3^n) + ((3 - sqrt(5))/2)^(3^n) - 1.
a(2*n+1) = ((3 + sqrt(5))/2)^(3^n) + ((3 - sqrt(5))/2)^(3^n) + 1. - Peter Bala, Nov 22 2012
EXAMPLE
From Peter Bala, Nov 22 2012: (Start)
Let x = (sqrt(5) - 1)/2. We have the alternating series expansions
x = 1 - 1/2 + 1/(2*4) - 1/(2*4*17) + 1/(2*4*17*19) - ...
x^2 = 1/2 - 1/(2*4) + 1/(2*4*17) - 1/(2*4*17*19) + ...
x^6 = 1/17 - 1/(17*19) + 1/(17*19*5777) - ...
as well as
x^3 = 1/4 - 1/(4*17) + 1/(4*17*19) - 1/(4*17*19*5777) + ...
4*x^9 = 1/19 - 1/(19*5777) + 1/(19*5777*5779) - ...
4*19*x^27 = 1/5779 - 1/(5779*192900153617) + ....
(End)
MATHEMATICA
Table[c=2*3^Floor[n/2]; 2*Fibonacci[c+1]-Fibonacci[c]-(-1)^n, {n, 0, 10}] (* Harvey P. Dale, Oct 22 2013 *)
PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[(3 - Sqrt[5])/2, 7!], 10] (* G. C. Greubel, Nov 14 2016 *)
PROG
(PARI) r=(1+sqrt(5))/2; for(n=1, 10, r=r/(r-floor(r)) print1(floor(r), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James A. Sellers, May 19 2000
STATUS
approved