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A219506 Pierce expansion of 2 - sqrt(3). 2
3, 5, 51, 53, 140451, 140453, 2770663499604051, 2770663499604053, 21269209556953516583554114034636483645584976451, 21269209556953516583554114034636483645584976453 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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 = 2 - sqrt(3).

Shallit has shown that the Pierce expansion of the quadratic irrational (c - sqrt(c^2 - 4))/2 has the form [c(0) - 1, c(0) + 1, c(1) - 1, c(1) + 1, c(2) - 1, c(2) + 1, ...], where c(0) = c and c(n+1) = c(n)^3 - 3*c(n). This is the case c = 4. For other cases see A006276 (c = 3), A219507 (c = 5) and A006275 (essentially c = 6 apart from the initial term).

The Pierce expansion of {(c - sqrt(c^2 - 4))/2}^(3^n) is [[c(n) - 1, c(n) + 1, c(n+1) - 1, c(n+1) + 1, c(n+2) - 1,c(n+2) + 1, ...].

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..13

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

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

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

EXAMPLE

We have the alternating series expansions

2 - sqrt(3) = 1/3 - 1/(3*5) + 1/(3*5*51) - 1/(3*5*51*53) + ...

(2 - sqrt(3))^3 = 1/51 - 1/(51*53) + 1/(51*53*140451) - ...

(2 - sqrt(3))^9 = 1/140451 - 1/(140451*140453) + ....

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[2 - Sqrt[3] , 7!], 10] (* G. C. Greubel, Nov 14 2016 *)

CROSSREFS

Cf. A006275, A006276, A112845, A219160, A219507, A219508.

Sequence in context: A120426 A077201 A196467 * A171775 A260227 A260226

Adjacent sequences:  A219503 A219504 A219505 * A219507 A219508 A219509

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Nov 22 2012

STATUS

approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)