A143515 Array D of denominators of Best Remaining Upper Approximates of x=(1+sqrt(5))/2, by antidiagonals.

1, 3, 2, 8, 4, 5, 21, 6, 7, 10, 55, 11, 9, 12, 13, 144, 16, 14, 17, 15, 18, 377, 29, 19, 22, 20, 23, 26, 987, 42, 24, 27, 25, 28, 31, 34, 2584, 76, 37, 32, 30, 33, 36, 39, 47, 6765, 110, 50, 45, 35, 38, 41, 44, 52, 60, 17711, 199, 63, 58, 40, 43, 46, 49, 57, 65

Offset: 1

Author

Clark Kimberling, Aug 22 2008

Keywords

• nonn
• tabl

Comments

(1) Row 1 of R consists of upper principal convergents to x.

(2) (row limits of R) = x; (column limits of R) = 0.

(3) Every positive integer occurs exactly once in D, so that as a sequence, this is a permutation of the positive integers.

(4) p=1+floor(q*r) for every p/q in R. Consequently, the terms of N are distinct and their ordered union is 1+A000201.

(5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is a principal convergent to x.

(6) Suppose n>=1 and p/q and s/t are consecutive terms in row n of R. Then (conjecture) p*t-q*s=n.

In general, for irrational r, let {n*r} denote the fractional part of n*r. Define t(1,1) = 1, and t(1,n) = least k such that {k*r} > {t(1,n-1)*x} for n >= 2. Inductively, for m >= 2 and n >= 1, let t(m,1) be the least k not already defined as a term in the array, and for n >= 2, define t(m,n) = least k such that {k*r} > {t(m,n-1)*x and k has not previously been defined as a term. Thus, every row of (t(m,n)) is strictly increasing. For r = (1+sqrt(5))/2, the array (t(m,n) is D. - Clark Kimberling, Feb 21 2021

Examples

			Northwest corner of D:
  1 3 8 21
  2 4 6 11
  5 7 9 14
  10 12 17 22
Northwest corner of R:
  2/1 5/3 13/8 34/21
  4/2 7/4 10/6 18/11
  9/5 12/7 15/9 23/14
  17/10 20/12 28/17 36/22
		

Links

• Clark Kimberling, Best lower and upper approximates to irrational numbers, Elemente der Mathematik, 52 (1997) 122-126.

Crossrefs

Cf. A000045, A000201, A143514, A143516.

Programs

Mathematica

r = N[(1 + Sqrt[5])/2, 100]; Table[d = 0; t[k] = {};
Do[a = FractionalPart[n*r];
  If[a > d && ! MemberQ[Apply[Union, Map[t[#] &, Range[k - 1]]], n],
   d = a; AppendTo[t[k], n]], {n, 10000}]; t[k], {k, 12}];
Column[Table[t[k], {k, 1, 12}]]
(* Peter J. C. Moses, Feb 18 2021 *)

Formula

For any positive irrational number x, define an array D by successive rows as follows: D(n,k) = least positive integer q not already in D such that there exists an integer p such that 0 < p/q - x < c/d - x for every positive rational number c/d that has 0 < d < q. Thus p/q is the "best remaining upper approximate" of x when all better upper approximates are unavailable. For each q, define N(n,k)=p and R(n,k)=p/q. Then R is the "array of best remaining upper approximates of x," D is the corresponding array of denominators and N, of numerators.