1,2

(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.

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 < 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.

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

Cf. A000045, A000201, A143515, A143516.

Sequence in context: A114744 A096114 A121664 this_sequence A085180 A114750 A145391

Adjacent sequences: A143511 A143512 A143513 this_sequence A143515 A143516 A143517

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Aug 22 2008, Aug 25 2008