A143527 Array D of denominators of Best Remaining Lower Approximates of x=sqrt(2), by antidiagonals.

1, 3, 2, 5, 4, 7, 17, 6, 9, 12, 29, 8, 11, 14, 19, 99, 10, 13, 16, 21, 24, 169, 22, 15, 18, 23, 26, 31, 577, 34, 27, 20, 25, 28, 33, 36, 985, 46, 39, 32, 37, 30, 35, 38, 41, 3363, 58, 51, 44, 49, 42, 47, 40, 43, 48, 5741, 128, 63, 56, 61, 54, 59, 52, 45, 50

Offset: 1

Author

Clark Kimberling, Aug 22 2008

Keywords

• nonn
• tabl

Comments

(1) Row 1 of R consists of the lower principal and lower 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, A143527 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 sequence A001951.

(5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is an upper 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.

Examples

			Northwest corner of D:
1 3 5 17
2 4 6 8
7 9 11 13
12 14 16 18
Northwest corner of R:
1/1 3/3 8/5 21/17
2/2 5/4 8/6 11/8
9/6 11/9 15/12 18/15
16/8 19/11 22/14 25/17
		

References

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

Crossrefs

Cf. A000129, A001951, A143514.

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 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.