%I #3 Mar 30 2012 18:57:10

%S 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,

%T 23,26,31,577,34,27,20,25,28,33,36,985,46,39,32,37,30,35,38,41,3363,

%U 58,51,44,49,42,47,40,43,48,5741,128,63,56,61,54,59,52,45,50

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

%C (1) Row 1 of R consists of the lower principal and lower intermediate convergents to x.

%C (2) (row limits of R) = x; (column limits of R) = 0.

%C (3) Every positive integer occurs exactly once in D, so that as a sequence, A143527 is a permutation of the positive integers.

%C (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.

%C (5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is an upper principal convergent to x.

%C (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.

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

%F 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.

%e Northwest corner of D:

%e 1 3 5 17

%e 2 4 6 8

%e 7 9 11 13

%e 12 14 16 18

%e Northwest corner of R:

%e 1/1 3/3 8/5 21/17

%e 2/2 5/4 8/6 11/8

%e 9/6 11/9 15/12 18/15

%e 16/8 19/11 22/14 25/17

%Y Cf. A000129, A001951, A143514.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Aug 22 2008