A143529 Array D of denominators of Best Remaining Approximates of x=sqrt(2), by antidiagonals.
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, 28, 35, 70, 24, 26, 38, 35, 30, 45, 47, 99, 34, 27, 39, 37, 40, 49, 52, 57, 169, 41, 31, 48, 43, 42, 50, 54, 76, 59, 408, 58, 36, 51, 55, 44, 62, 69, 81, 88
Offset: 1
Examples
Northwest corner of D: 1 2 3 5 4 6 7 10 8 9 13 14 11 16 20 32 Northwest corner of R: 1/1 3/2 4/3 7/5 6/4 8/6 10/7 14/10 11/8 13/9 18/13 20/14 16/11 23/16 28/20 45/32
Formula
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.
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