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A143516 Array D of denominators of Best Remaining Approximates of x=(1+sqrt(5))/2, by antidiagonals. 4

%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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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)