A143516 -id:A143516 - cp's OEIS frontend

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

Clark Kimberling, Aug 22 2008, Aug 25 2008

(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

			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

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

Cf. A000045, A000201, A143515, A143516.

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 *)

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.

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

Original entry on oeis.org

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

Clark Kimberling, Aug 22 2008

(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

			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

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

Cf. A000045, A000201, A143514, A143516.

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 *)

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.

A143529 Array D of denominators of Best Remaining Approximates of x=sqrt(2), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 5, 7, 9, 11, 12, 10, 13, 16, 18, 17, 19, 14, 20, 21, 23, 29, 22, 15, 32, 25, 28, 35, 70, 24, 26, 38, 35, 30, 45, 47, 99, 34, 27, 39, 37, 40, 49, 52, 57, 169, 41, 31, 48, 43, 42, 50, 54, 76, 59, 408, 58, 36, 51, 55, 44, 62, 69, 81, 88

Offset: 1

Clark Kimberling, Aug 23 2008

(1) Row 1 of R consists of principal and intermediate convergents to x; however, not all intermediate convergents occur; e.g., 10/7, 58/41, 338/239 are missing.
(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, A143529 is a permutation of the positive integers.

			Northwest corner of D:
1 2 3 5
4 6 7 10
8 9 13 14
11 16 20 32
Northwest corner of R:
1/1 3/2 4/3 7/5
6/4 8/6 10/7 14/10
11/8 13/9 18/13 20/14
16/11 23/16 28/20 45/32

Cf. A143516, A143527, A143528.

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 approximate" of x when all better 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 approximates of x," D is the corresponding array of denominators and N, of numerators.

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

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

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A143529 Array D of denominators of Best Remaining Approximates of x=sqrt(2), by antidiagonals.

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