The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A006275 Pierce expansion of sqrt(2) - 1. (Formerly M1342) 10
 2, 5, 7, 197, 199, 7761797, 7761799, 467613464999866416197, 467613464999866416199, 102249460387306384473056172738577521087843948916391508591105797 (list; graph; refs; listen; history; text; internal format)
 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 - #[] #[]]], Expand[1 - #[] #[]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Sqrt - 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 EXTENSIONS More terms from James A. Sellers, May 19 2000 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 27 05:24 EDT 2021. Contains 346305 sequences. (Running on oeis4.)