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A006275 Pierce expansion of sqrt(2) - 1.

%I A006275 M1342 #58 Jul 02 2025 16:01:54
%S A006275 2,5,7,197,199,7761797,7761799,467613464999866416197,
%T A006275 467613464999866416199,
%U A006275 102249460387306384473056172738577521087843948916391508591105797
%N A006275 Pierce expansion of sqrt(2) - 1.
%C A006275 From _Peter Bala_, Nov 22 2012: (Start)
%C A006275 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 A006275 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 A006275 (End)
%D A006275 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006275 G. C. Greubel, <a href="/A006275/b006275.txt">Table of n, a(n) for n = 0..14</a>
%H A006275 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 A006275 Jeffrey Shallit, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/22-4/shallit1.pdf">Some predictable Pierce expansions</a>, Fib. Quart., 22 (1984), 332-335.
%H A006275 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PierceExpansion.html">Pierce Expansion</a>.
%F A006275 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 A006275 From _Peter Bala_, Nov 22 2012: (Start)
%F A006275 a(2*n+2) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) + 1.
%F A006275 a(2*n+1) = (3 + 2*sqrt(2))^(3^n) + (3 - 2*sqrt(2))^(3^n) - 1. (End)
%F A006275 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 A006275 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 A006275 From _Peter Bala_, Nov 22 2012: (Start)
%e A006275 Let x = sqrt(2) - 1. We have the alternating series expansions
%e A006275 x = 1/2 - 1/(2*5) + 1/(2*5*7) - 1/(2*5*7*197) + ...
%e A006275 x^3 = 1/14 - 1/(14*197) + 1/(14*197*199) - ...
%e A006275 x^9 = 1/2786 - 1/(2786*7761797) + 1/(2786*7761797*7761799) - ...,
%e A006275 where 2786 = 2*7*199, and also
%e A006275 x^2 = 1/5 - 1/(5*7) + 1/(5*7*197) - 1/(5*7*197*199) + ...
%e A006275 x^6 = 1/197 - 1/(197*199) + 1/(197*199*7761797) - ...
%e A006275 x^18 = 1/7761797 - 1/(7761797*7761799) + ....
%e A006275 (End)
%t A006275 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 A006275 (PARI) my(r=1+quadgen(8)); for(n=1, 10, print1(floor(r), ", "); r=r/(r-floor(r)));
%Y A006275 Cf. A014176, A006276.
%K A006275 nonn,easy
%O A006275 0,1
%A A006275 _N. J. A. Sloane_
%E A006275 More terms from _James Sellers_, May 19 2000