A394037 Rectangular array, read by descending antidiagonals: denominators in the best approximating array to the golden ratio, phi (A001622). See Comments.

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

Offset

1,2

Comments

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:

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

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

(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).

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

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.

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).

Every row and every column converges to x.

Links

Table of n, a(n) for n=1..78.

Example

Corner of approximating array:
    2/1     3/2     5/3     8/5    13/8    21/13   34/21   55/34   89/55
    1/1     4/3     7/4    11/7    18/11   29/18   47/29   76/47  123/76
    9/5    12/7    14/9    19/12   23/14   31/19   37/23   50/31   60/37
   17/10   17/11   25/16   27/17   35/22   41/25   44/27   45/28   57/35
   22/13   28/17   43/27   49/30   53/33   70/43   79/49   86/53   92/57
   30/19   33/20   46/29   51/31   51/32   59/36   61/38   75/46   82/51
   38/23   59/37   64/39   67/41   69/43   83/51   96/59  103/64  108/67
   43/26   67/42   77/47   77/48   98/61  101/62  109/67  122/75  124/77
Corner of array of denominators:
   1    2    3    5    8   13   21   34   55
   1    3    4    7   11   18   29   47   76
   5    7    9   12   14   19   23   31   37
  10   11   16   17   22   25   27   28   35
  13   17   27   30   33   43   49   53   57
  19   20   29   31   32   36   38   46   51
  23   37   39   41   43   51   59   64   67
  26   42   47   48   61   62   67   75   77
Corner of array of approximations to best approximating array:
  2.      1.5     1.66667 1.6     1.625   1.61538
  1.      1.33333 1.75    1.57143 1.63636 1.61111
  1.8     1.71429 1.55556 1.58333 1.64286 1.63158
  1.7     1.54545 1.5625  1.58824 1.59091 1.64
  1.69231 1.64706 1.59259 1.63333 1.60606 1.62791
  1.57895 1.65    1.58621 1.64516 1.59375 1.63889
  1.65217 1.59459 1.64103 1.63415 1.60465 1.62745
  1.65385 1.59524 1.6383  1.60417 1.60656 1.62903
  1.63265 1.60377 1.63077 1.60563 1.63014 1.60811

Crossrefs

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

Sequence in context: A288005 A237832 A074500 * A107237 A070047 A101198

Adjacent sequences: A394034 A394035 A394036 * A394038 A394039 A394040

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Mar 27 2026

Status

approved