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

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

1,2

Comments

(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.

Links

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

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 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.

Example

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

Crossrefs

Cf. A143516, A143527, A143528.

Sequence in context: A342793 A178898 A296616 * A331010 A261351 A249054

Adjacent sequences: A143526 A143527 A143528 * A143530 A143531 A143532

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 23 2008

Status

approved