A143527 -id:A143527 - cp's OEIS frontend

A143528 Array D of denominators of Best Remaining Upper Approximates of x=sqrt(2), by antidiagonals.

1, 2, 3, 7, 4, 5, 12, 9, 6, 8, 41, 14, 11, 13, 10, 70, 19, 16, 18, 15, 17, 239, 24, 21, 23, 20, 22, 29, 408, 53, 26, 28, 25, 27, 34, 46, 1393, 82, 31, 33, 30, 32, 39, 51, 58, 2378, 111, 36, 38, 35, 37, 44, 56, 63, 75, 8119, 140, 65, 43, 40, 42, 49, 61, 68, 80

Offset: 1

Clark Kimberling, Aug 23 2008

(1) Row 1 of R consists of the upper principal and upper intermediate 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, A143528 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 the sequence 1+A001951.
(5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is a lower 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.

			Northwest corner of D:
1 2 7 12
3 4 9 14
5 6 11 16
8 13 18 23
Northwest corner of R:
2/1 3/2 10/7 17/12
5/3 6/4 13/9 20/14
8/5 9/6 16/11 23/16
12/8 19/13 26/18 33/23

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

Cf. A001951, A084068, A143515, A143527, A143529.

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.

cp's OEIS Frontend

A143528 Array D of denominators of Best Remaining Upper Approximates of x=sqrt(2), by antidiagonals.

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