A143528 - OEIS

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

%S 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,

%T 20,22,29,408,53,26,28,25,27,34,46,1393,82,31,33,30,32,39,51,58,2378,

%U 111,36,38,35,37,44,56,63,75,8119,140,65,43,40,42,49,61,68,80

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

%C (1) Row 1 of R consists of the upper principal and upper 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, A143528 is a permutation of the positive integers.

%C (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.

%C (5) Conjecture: Every (N(n,k+1)-N(n,k))/(D(n,k+1)-D(n,k)) is a lower 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) p*t-q*s=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 < 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.

%e Northwest corner of D:

%e 1 2 7 12

%e 3 4 9 14

%e 5 6 11 16

%e 8 13 18 23

%e Northwest corner of R:

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

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

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

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

%Y Cf. A001951, A084068, A143515, A143527, A143529.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Aug 23 2008