%I A143527
%S A143527 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,
%T A143527 26,31,577,34,27,20,25,28,33,36,985,46,39,32,37,30,35,38,41,3363,58,51,
%U A143527 44,49,42,47,40,43,48,5741,128,63,56,61,54,59,52,45,50
%N A143527 Array D of denominators of Best Remaining Lower Approximates of x=sqrt(2), 
               by antidiagonals.
%C A143527 (1) Row 1 of R consists of the lower principal and lower intermediate 
               convergents to x.
%C A143527 (2) (row limits of R) = x; (column limits of R) = 0.
%C A143527 (3) Every positive integer occurs exactly once in D, so that as a sequence, 
               A143527 is a permutation of the positive integers.
%C A143527 (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 A143527 (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 A143527 (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 A143527 C. Kimberling, "Best lower and upper approximates to irrational numbers,
               " Elemente der Mathematik 52 (1997) 122-126.
%F A143527 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 A143527 Northwest corner of D:
%e A143527 1 3 5 17
%e A143527 2 4 6 8
%e A143527 7 9 11 13
%e A143527 12 14 16 18
%e A143527 Northwest corner of R:
%e A143527 1/1 3/3 8/5 21/17
%e A143527 2/2 5/4 8/6 11/8
%e A143527 9/6 11/9 15/12 18/15
%e A143527 16/8 19/11 22/14 25/17
%Y A143527 Cf. A000129, A001951, A143514.
%Y A143527 Sequence in context: A138310 A137707 A164380 this_sequence A095720 A115297 
               A054430
%Y A143527 Adjacent sequences: A143524 A143525 A143526 this_sequence A143528 A143529 
               A143530
%K A143527 nonn,tabl
%O A143527 1,2
%A A143527 Clark Kimberling (ck6(AT)evansville.edu), Aug 22 2008