A098457 Farey Bisection Expansion of sqrt(7).

1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1

Offset

1,1

Comments

We define the Farey Bisection Expansion (FBE) of the nonnegative real number x to be the sequence {a(n)} of 0's and 1's determined as follows. Set na(0)=0, da(0)=1, nb(0)=1 and db(0)=0. For n=1, 2, 3,..., set num=na(n-1)+nb(n-1) and den=da(n-1)+db(n-1); if x<n/b, set a(n)=0, na(n)=na(n-1), da(n)=da(n-1), nb(n)=num, db(n)=den, else set a(n)=1, na(n)=num, da(n)=den, nb(n)=nb(n-1), db(n)=db(n-1). (The process is akin to that of locating the zero of a function by the bisection method, simply recording which successive subinterval, the left or the right, the zero lies at each refinement.) The FBE of Sqrt[7] is periodic with period 7. The RUNS transform of FBE(x) is the sequence of partial quotients of the continued fraction of x. As can be seen, RUNS(FBE(Sqrt[7]))={2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1,...}, which is A010121.

Links

Table of n, a(n) for n=1..105.

N. J. A. Sloane, Stern-Brocot or Farey Tree

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,1).

Formula

From Wesley Ivan Hurt, Jul 11 2016: (Start)

G.f.: x * (1 + x + x^3 + x^5 + x^6) / (1 - x^7).

a(n) = a(n-7) for n>7.

a(n) = 1 - Sum_{k=1..4} floor((n + k)/7)*(-1)^k. (End)

a(n+1) = (-1)^(mod(mod(n, 7), 3)>0) * A131372(n). - Michael Somos, Dec 26 2016

Example

G.f. = x + x^2 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^13 + x^14 + x^15 + ...

Maple

seq(op([1, 1, 0, 1, 0, 1, 1]), n=0..20); # Wesley Ivan Hurt, Jul 11 2016

Mathematica

LinearRecurrence[{0, 0, 0, 0, 0, 0, 1}, {1, 1, 0, 1, 0, 1, 1}, 105] (* Ray Chandler, Aug 26 2015 *)

Prog

(Magma) &cat [[1, 1, 0, 1, 0, 1, 1]^^20]; // Wesley Ivan Hurt, Jul 11 2016
(PARI) {a(n) = [1, 1, 0, 1, 0, 1, 1][(n-1)%7+1]}; /* Michael Somos, Dec 26 2016 */

Crossrefs

Cf. A010121, A097853, A098458, A131372.

Sequence in context: A267579 A285414 A131372 * A284793 A260397 A137161

Adjacent sequences: A098454 A098455 A098456 * A098458 A098459 A098460

Keyword

nonn,easy

Author

John W. Layman, Sep 08 2004

Status

approved