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A143515 Array D of denominators of Best Remaining Upper Approximates of x=(1+sqrt(5))/2, by antidiagonals. 3
1, 3, 2, 8, 4, 5, 21, 6, 7, 10, 55, 11, 9, 12, 13, 144, 16, 14, 17, 15, 18, 377, 29, 19, 22, 20, 23, 26, 987, 42, 24, 27, 25, 28, 31, 34, 2584, 76, 37, 32, 30, 33, 36, 39, 47, 6765, 110, 50, 45, 35, 38, 41, 44, 52, 60, 17711, 199, 63, 58, 40, 43, 46, 49, 57, 65 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

(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.

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 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

CROSSREFS

Cf. A000045, A000201, A143514, A143516.

Sequence in context: A110938 A135852 A191731 * A082333 A162728 A127300

Adjacent sequences:  A143512 A143513 A143514 * A143516 A143517 A143518

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 22 2008

STATUS

approved

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Last modified July 27 20:09 EDT 2017. Contains 289866 sequences.