%I #5 Dec 07 2016 10:31:33
%S 1,2,4,3,6,9,5,7,11,12,8,10,15,14,17,13,16,19,20,22,33,21,18,23,27,25,
%T 38,34,26,24,28,30,40,41,43,55,42,29,31,32,49,48,46,51,89,47,39,36,53,
%U 54,56,59,72,144,68,50,52,44,66,61,62,64,77
%N Array D of denominators of Best Remaining Approximates of x=(1+sqrt(5))/2, by antidiagonals.
%C (1) Row 1 of R consists of principal 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, A143516 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 9 11 15 19
%e 12 14 20 27
%e Northwest corner of R:
%e 2/1 3/2 5/3 8/5
%e 6/4 10/6 11/7 16/10
%e 15/9 18/11 24/15 31/19
%e 19/12 23/14 32/20 44/27
%Y Cf. A000045, A143514, A143515.
%K nonn,tabl,frac
%O 1,2
%A _Clark Kimberling_, Aug 22 2008
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