A006275 Pierce expansion of sqrt(2) - 1. (Formerly M1342)

2, 5, 7, 197, 199, 7761797, 7761799, 467613464999866416197, 467613464999866416199, 102249460387306384473056172738577521087843948916391508591105797

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

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.

(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..14

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 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

From Peter Bala, Nov 22 2012: (Start)

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

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

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

Example

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, ...
From Peter Bala, Nov 22 2012: (Start)
Let x = sqrt(2) - 1. We have the alternating series expansions
x = 1/2 - 1/(2*5) + 1/(2*5*7) - 1/(2*5*7*197) + ...
x^3 = 1/14 - 1/(14*197) + 1/(14*197*199) - ...
x^9 = 1/2786 - 1/(2786*7761797) + 1/(2786*7761797*7761799) - ...,
where 2786 = 2*7*199, and also
x^2 = 1/5 - 1/(5*7) + 1/(5*7*197) - 1/(5*7*197*199) + ...
x^6 = 1/197 - 1/(197*199) + 1/(197*199*7761797) - ...
x^18 = 1/7761797 - 1/(7761797*7761799) + ....
(End)

Mathematica

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 *)

Prog

(PARI) r=1+sqrt(2); for(n=1, 10, r=r/(r-floor(r)); print1(floor(r), ", "))

Crossrefs

Cf. A006276.

Sequence in context: A041961 A242169 A058854 * A042673 A214705 A252283

Adjacent sequences: A006272 A006273 A006274 * A006276 A006277 A006278

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 19 2000

Status

approved