

A219508


Pierce expansion of 4*(3  2*sqrt(2)).


6



1, 3, 16, 17, 72, 577, 2312, 665857, 2663432, 886731088897, 3546924355592, 1572584048032918633353217, 6290336192131674533412872, 4946041176255201878775086487573351061418968498177
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OFFSET

0,2


COMMENTS

For x in the open interval (0,1) define the map f(x) = 1  x*floor(1/x). The nth term (n >= 0) in the Pierce expansion of x is given by floor(1/f^(n)(x)), where f^(n)(x) denotes the nth iterate of the map f, with the convention that f^(0)(x) = x.
Paradis et al. have determined the Pierce expansion of the quadratic irrationality 2*(p  1)*(p  sqrt(p^2  1)), p a positive integer greater than or equal to 3. The present sequence is the case p = 3. For other cases see A219509 (p = 5), A219510 (p = 7) and A219511 (p = 9).
Compare this Pierce expansion for 4*(3  2*sqrt(2)), with terms determined by quadratic recurrences, with the Pierce expansion of 3  2*sqrt(2) given in A006275, where the terms are determined by cubic recurrences.


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..22
J. Paradis, P. Viader, L. Bibiloni Approximation to quadratic irrationals and their Pierce expansions, The Fibonacci Quarterly, Vol.36 No. 2 (1998) 146153.
T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly, Vol. 36 No. 10, (1929) p.523525.
Eric Weisstein's World of Mathematics, Pierce Expansion


FORMULA

a(2*n) = 2*{(1 + sqrt(2))^(2^n) + (1  sqrt(2))^(2^n) + 2} for n >= 1.
a(2*n1) = 1/2*{(1 + sqrt(2))^(2^n) + (1  sqrt(2))^(2^n)} for n >= 1.
Recurrence equations: a(0) = 1, a(1) = 3 and for n >= 1, a(2*n) = 4*(a(2*n1) + 1) and a(2*n+1) = 2*{a(2*n1)}^2  1.
It follows that a(2*n) = 8*a(2*n3)^2 for n >=2.
4*(3  2*sqrt(2)) = sum {n >= 0} 1/product {k = 0..n} a(k) = 1  1/3 + 1/(3*16)  1/(3*16*17) + 1/(3*16*17*72)  ....
a(2*n) = 8*A001601(n1)^2 for n >= 2
a(2*n1) = A001601(n) for n >= 1.


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[4*(3  2*Sqrt[2]) , 7!], 10] (* G. C. Greubel, Nov 14 2016 *)


PROG

(PARI) r=(3 + 2*sqrt(2))/4; for(n=1, 10, print(floor(r), ", "); r=r/(rfloor(r))) \\ G. C. Greubel, Nov 15 2016


CROSSREFS

Cf. A001601, A006275, A219509 (p = 5), A219510 (p = 7), A219511 (p = 9).
Sequence in context: A272329 A076623 A068516 * A032922 A103655 A022126
Adjacent sequences: A219505 A219506 A219507 * A219509 A219510 A219511


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Nov 23 2012


STATUS

approved



