1,2

(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.

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.

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.

Sequence in context: A006016 A054239 A048680 this_sequence A103867 A075375 A065562

Adjacent sequences: A143526 A143527 A143528 this_sequence A143530 A143531 A143532

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Aug 23 2008