|
%I #3 Mar 30 2012 18:57:10
%S 1,2,4,3,6,8,5,7,9,11,12,10,13,16,18,17,19,14,20,21,23,29,22,15,32,25,
%T 28,35,70,24,26,38,35,30,45,47,99,34,27,39,37,40,49,52,57,169,41,31,
%U 48,43,42,50,54,76,59,408,58,36,51,55,44,62,69,81,88
%N Array D of denominators of Best Remaining Approximates of x=sqrt(2), by antidiagonals.
%C (1) Row 1 of R consists of principal and intermediate convergents to x; however, not all intermediate convergents occur; e.g., 10/7, 58/41, 338/239 are missing.
%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, A143529 is a permutation of the positive integers.
%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 approximate" of x when all better 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 approximates of x," D is the corresponding array of denominators and N, of numerators.
%e Northwest corner of D:
%e 1 2 3 5
%e 4 6 7 10
%e 8 9 13 14
%e 11 16 20 32
%e Northwest corner of R:
%e 1/1 3/2 4/3 7/5
%e 6/4 8/6 10/7 14/10
%e 11/8 13/9 18/13 20/14
%e 16/11 23/16 28/20 45/32
%Y Cf. A143516, A143527, A143528.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Aug 23 2008
|
