A394037 - OEIS

%I #14 Apr 11 2026 17:49:39

%S 1,2,1,3,3,5,5,4,7,10,8,7,9,11,13,13,11,12,16,17,19,21,18,14,17,27,20,

%T 23,34,29,19,22,30,29,37,26,55,47,23,25,33,31,39,42,49,89,76,31,27,43,

%U 32,41,47,53,52,144,123,37,28,49,36,43,48,65,58,63,233,199,50,35,53,38,51,61,71,68,79,89

%N Rectangular array, read by descending antidiagonals: denominators in the best approximating array to the golden ratio, phi (A001622). See Comments.

%C Let Q = {a/b: gcd(a,b)=1, a>0, b>0}; i.e., the positive rationals, reduced to lowest terms, and let x be a positive irrational number. For any dense subset S of Q, a sequence of fractions a(n)/b(n) in S is the S-best approximating sequence if it satisfies these three conditions for all k>=1:

%C   (1) |a(k)/b(k) - x| > |a(k+1)/b(k+1) - x|

%C   (2) b(k) < b(k+1)

%C   (3) if b(k) < v < b(k+1) for some u/v in S, then u/v is not between a(k)/b(k) and a(k+1)/b(k+1).

%C Define the best approximating array to x by rows, r(n,k), for k>=1, as follows:

%C Row 1: With S = Q, put r(1,1) = a(1)/b(1), where a(1) = round(x) and b(1)=1. The terms r(1,k) are then uniquely determined by conditions (1), (2), (3). Note that row 1 consists of the convergents to x.

%C Rows n>=2: With S = (Q after deleting all fractions in rows 1,2,...,n-1), let r(n,1) = the fraction a/b in S that is nearest x and has least denominator b that is new; i.e., a/b is not in the first n-1 rows. The terms of row (r(n,k)) are uniquely determined by (1), (2), (3).

%C Every row and every column converges to x.

%e Corner of approximating array:

%e     2/1     3/2     5/3     8/5    13/8    21/13   34/21   55/34   89/55

%e     1/1     4/3     7/4    11/7    18/11   29/18   47/29   76/47  123/76

%e     9/5    12/7    14/9    19/12   23/14   31/19   37/23   50/31   60/37

%e    17/10   17/11   25/16   27/17   35/22   41/25   44/27   45/28   57/35

%e    22/13   28/17   43/27   49/30   53/33   70/43   79/49   86/53   92/57

%e    30/19   33/20   46/29   51/31   51/32   59/36   61/38   75/46   82/51

%e    38/23   59/37   64/39   67/41   69/43   83/51   96/59  103/64  108/67

%e    43/26   67/42   77/47   77/48   98/61  101/62  109/67  122/75  124/77

%e Corner of array of denominators:

%e    1    2    3    5    8   13   21   34   55

%e    1    3    4    7   11   18   29   47   76

%e    5    7    9   12   14   19   23   31   37

%e   10   11   16   17   22   25   27   28   35

%e   13   17   27   30   33   43   49   53   57

%e   19   20   29   31   32   36   38   46   51

%e   23   37   39   41   43   51   59   64   67

%e   26   42   47   48   61   62   67   75   77

%e Corner of array of approximations to best approximating array:

%e   2.      1.5     1.66667 1.6     1.625   1.61538

%e   1.      1.33333 1.75    1.57143 1.63636 1.61111

%e   1.8     1.71429 1.55556 1.58333 1.64286 1.63158

%e   1.7     1.54545 1.5625  1.58824 1.59091 1.64

%e   1.69231 1.64706 1.59259 1.63333 1.60606 1.62791

%e   1.57895 1.65    1.58621 1.64516 1.59375 1.63889

%e   1.65217 1.59459 1.64103 1.63415 1.60465 1.62745

%e   1.65385 1.59524 1.6383  1.60417 1.60656 1.62903

%e   1.63265 1.60377 1.63077 1.60563 1.63014 1.60811

%Y Cf. A001622, A000045 (row 1), A000032 (row 2), A394033, A394647.

%K nonn,tabl,frac

%O 1,2

%A _Clark Kimberling_, Mar 27 2026