A143514 Array D of denominators of Best Remaining Lower Approximates of x=(1+sqrt(5))/2, by antidiagonals.

1, 2, 3, 5, 4, 6, 13, 7, 9, 8, 34, 10, 12, 11, 16, 89, 18, 15, 14, 19, 21, 233, 26, 23, 17, 22, 24, 29, 610, 47, 31, 20, 25, 27, 32, 37, 1597, 68, 39, 28, 33, 30, 35, 40, 42, 4181, 123, 60, 36, 41, 38, 43, 48, 45, 50, 10946, 178, 81, 44, 49, 46, 51, 56, 53, 58

Offset

1,2

Comments

(1) Row 1 of R consists of lower 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, A143514 is a permutation of the positive integers.

(4) p=floor(q*r) for every p/q in R. Consequently, the terms of N are distinct and their ordered union is the lower Wythoff sequence, 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) q*s-p*t=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 decreasing. For r = (1+sqrt(5))/2, the array (t(m,n)) is D. - Clark Kimberling, Feb 21 2021

Links

Table of n, a(n) for n=1..65.

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

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 < x - p/q < x - c/d for every positive rational number c/d that has 0 < d < q. Thus p/q is the "best remaining lower approximate" of x when all better lower 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 lower approximates of x," D is the corresponding array of denominators and N, of numerators.

Example

Northwest corner of D:
  1 2 5 13
  3 4 7 10
  6 9 12 15
  8 11 14 17
Northwest corner of R:
  1/1 3/2 8/5 21/13
  4/3 6/4 11/7 16/10
  9/6 14/9 19/12 24/15
  12/8 17/11 22/14 27/17

Mathematica

r = N[(1 + Sqrt[5])/2, 100]; Table[d = 1; 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 *)

Crossrefs

Cf. A000045, A000201, A143515, A143516.

Sequence in context: A344169 A121664 A352943 * A374801 A085180 A360209

Adjacent sequences: A143511 A143512 A143513 * A143515 A143516 A143517

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Aug 22 2008, Aug 25 2008

Status

approved