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

Author

John W. Layman, Sep 08 2004

Keywords

• nonn
• easy

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 xA010121.

Examples

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

Links

• 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).

Crossrefs

Cf. A010121, A097853, A098458, A131372.

Programs

Magma

&cat [[1, 1, 0, 1, 0, 1, 1]^^20]; // Wesley Ivan Hurt, Jul 11 2016

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 *)

PARI

{a(n) = [1, 1, 0, 1, 0, 1, 1][(n-1)%7+1]}; /* Michael Somos, Dec 26 2016 */

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