1,2

(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.

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

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.

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

Cf. A000045, A000201, A143514, A143516.

Sequence in context: A144189 A110938 A135852 this_sequence A082333 A162728 A127300

Adjacent sequences: A143512 A143513 A143514 this_sequence A143516 A143517 A143518

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Aug 22 2008