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A143514 Array D of denominators of Best Remaining Lower Approximates of x=(1+sqrt(5))/2, by antidiagonals. 4
1, 2, 3, 5, 4, 6, 13, 7, 9, 8, 34, 10, 12, 11, 16, 89, 18, 15, 14, 19, 21, 233, 26, 23, 17, 22, 24, 29, 610, 47, 31, 20, 25, 27, 32, 37, 1597, 68, 39, 28, 33, 30, 35, 40, 42, 4181, 123, 60, 36, 41, 38, 43, 48, 45, 50, 10946, 178, 81, 44, 49, 46, 51, 56, 53, 58 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

(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.

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 < 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.

EXAMPLE

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

CROSSREFS

Cf. A000045, A000201, A143515, A143516.

Sequence in context: A114744 A096114 A121664 * A085180 A114750 A234923

Adjacent sequences:  A143511 A143512 A143513 * A143515 A143516 A143517

KEYWORD

nonn,tabl,frac

AUTHOR

Clark Kimberling, Aug 22 2008, Aug 25 2008

STATUS

approved

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Last modified May 29 21:28 EDT 2017. Contains 287257 sequences.