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

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, 38, 34, 26, 24, 28, 30, 40, 41, 43, 55, 42, 29, 31, 32, 49, 48, 46, 51, 89, 47, 39, 36, 53, 54, 56, 59, 72, 144, 68, 50, 52, 44, 66, 61, 62, 64, 77

Offset: 1

Author

Clark Kimberling, Aug 22 2008

Keywords

• nonn
• tabl
• frac

Comments

(1) Row 1 of R consists of principal convergents to x.

(2) (row limits of R) = x; (column limits of R) = 0.

(3) Every positive integer occurs exactly once in D, so that as a sequence, A143516 is a permutation of the positive integers.

Examples

			Northwest corner of D:
1 2 3 5
4 6 7 10
9 11 15 19
12 14 20 27
Northwest corner of R:
2/1 3/2 5/3 8/5
6/4 10/6 11/7 16/10
15/9 18/11 24/15 31/19
19/12 23/14 32/20 44/27
		

Crossrefs

Cf. A000045, A143514, A143515.

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.