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Pierce expansion of sqrt(2) - 1.
(Formerly M1342)
12

%I M1342 #54 Aug 02 2024 09:57:52

%S 2,5,7,197,199,7761797,7761799,467613464999866416197,

%T 467613464999866416199,

%U 102249460387306384473056172738577521087843948916391508591105797

%N Pierce expansion of sqrt(2) - 1.

%C From _Peter Bala_, Nov 22 2012: (Start)

%C 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. The present sequence is the case x = sqrt(2) - 1.

%C The Pierce expansion of (sqrt(2) - 1)^(3^n) is [a(0)*a(2)*...*a(2*n), a(2*n+1), a(2*n+2), ...] = [sqrt(a(2*n+1) - 1), a(2*n+1), a(2*n+2), ...]. The Pierce expansion of (sqrt(2) - 1)^(2*3^n) is [a(2*n+1), a(2*n+2), ...]. Some examples of the associated alternating series are given below.

%C (End)

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H G. C. Greubel, <a href="/A006275/b006275.txt">Table of n, a(n) for n = 0..14</a>

%H T. A. Pierce, <a href="http://www.jstor.org/stable/2299963">On an algorithm and its use in approximating roots of algebraic equations</a>, Amer. Math. Monthly, Vol. 36, No. 10 (1929), pp. 523-525.

%H Jeffrey Shallit, <a href="http://www.fq.math.ca/Scanned/22-4/shallit1.pdf">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.

%F Let u(0)=1+sqrt(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

%F From _Peter Bala_, Nov 22 2012: (Start)

%F a(2*n+2) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) + 1.

%F a(2*n+1) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) - 1. (End)

%F sqrt(2) - 1 = a(0)/a(1) + (a(0)*a(2))/(a(1)*a(3)) + (a(0)*a(2)*a(4))/(a(1)*a(3)*a(5)) + ... = 2/5 + (2*7)/(5*197) + (2*7*199)/(5*197*7761797) + ... . - _Peter Bala_, Dec 03 2012

%e Let c(0)=6, c(n+1) = c(n)^3-3*c(n); then this sequence is 2, c(0)-1, c(0)+1, c(1)-1, c(1)+1, c(2)-1, c(2)+1, ...

%e From _Peter Bala_, Nov 22 2012: (Start)

%e Let x = sqrt(2) - 1. We have the alternating series expansions

%e x = 1/2 - 1/(2*5) + 1/(2*5*7) - 1/(2*5*7*197) + ...

%e x^3 = 1/14 - 1/(14*197) + 1/(14*197*199) - ...

%e x^9 = 1/2786 - 1/(2786*7761797) + 1/(2786*7761797*7761799) - ...,

%e where 2786 = 2*7*199, and also

%e x^2 = 1/5 - 1/(5*7) + 1/(5*7*197) - 1/(5*7*197*199) + ...

%e x^6 = 1/197 - 1/(197*199) + 1/(197*199*7761797) - ...

%e x^18 = 1/7761797 - 1/(7761797*7761799) + ....

%e (End)

%t 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[Sqrt[2] - 1, 7!], 10] (* _G. C. Greubel_, Nov 14 2016 *)

%o (PARI) my(r=1+quadgen(8)); for(n=1, 10, print1(floor(r), ", "); r=r/(r-floor(r)));

%Y Cf. A014176, A006276.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, May 19 2000