A143528 - OEIS

A143528

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

1, 2, 3, 7, 4, 5, 12, 9, 6, 8, 41, 14, 11, 13, 10, 70, 19, 16, 18, 15, 17, 239, 24, 21, 23, 20, 22, 29, 408, 53, 26, 28, 25, 27, 34, 46, 1393, 82, 31, 33, 30, 32, 39, 51, 58, 2378, 111, 36, 38, 35, 37, 44, 56, 63, 75, 8119, 140, 65, 43, 40, 42, 49, 61, 68, 80

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OFFSET

1,2

COMMENTS

(1) Row 1 of R consists of the upper principal and upper 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, A143528 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 the sequence 1+A001951.

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

REFERENCES

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

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 < 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.

EXAMPLE

Northwest corner of D:

1 2 7 12

3 4 9 14

5 6 11 16

8 13 18 23

Northwest corner of R:

2/1 3/2 10/7 17/12

5/3 6/4 13/9 20/14

8/5 9/6 16/11 23/16

12/8 19/13 26/18 33/23

CROSSREFS