A394033 Rectangular array, read by descending antidiagonals: denominators in the best lower approximating array to the golden ratio, phi (A001622). (See Comments).

1, 2, 3, 5, 7, 9, 13, 12, 16, 11, 34, 17, 19, 27, 29, 89, 18, 22, 28, 32, 37, 233, 31, 23, 51, 33, 38, 42, 610, 44, 41, 59, 61, 71, 43, 48, 1597, 47, 49, 67, 64, 74, 79, 87, 53, 4181, 81, 57, 70, 75, 77, 85, 95, 97, 58, 10946, 115, 60, 73, 83, 80, 101, 103

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 lower approximating sequence if it satisfies these three conditions:

(1) a(1)/b(1) < a(2)/b(2) < a(3)/b(3) < ... < x

(2) b(1) < b(2) < b(3) < ...

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

Define the best lower 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) = floor(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 lower 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 < 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. Regarding the arrays of denominators and numerators for the lower and upper best approximating arrays, every row and every column is strictly increasing.

For the present array, x = phi.

Links

Table of n, a(n) for n=1..63.

Example

Corner of array of denominators:
   1     2    5   13   34   89  233  610  1597  4181  10946  28657
   3     7   12   17   18   31   44   47    81   115    123    212
   9    16   19   22   23   41   49   57    60   107    128    149
   11   27   28   51   59   67   70   73   133   154    157    280
   29   32   33   61   64   75   83   86   159   167    175    183
   37   38   71   74   77   80   96   99   185   191    301    322
Corner of array of numerators:
    1    3    8   21    55   144   377   987  2584  6765  17711
    4   11   19   27    29    50    71    76   131   186    199
   14   25   30   35    37    66    79    92    97   173    207
   17   43   45   82    95   108   113   118   215   249    254
   46   51   53   98   103   121   134   139   257   270    283
   59   61  114  119   124   129   155   160   299   309    487
Corner of array of approximations:
   1.        1.5       1.6       1.61538   1.61765
   1.33333   1.57143   1.58333   1.58824   1.61111
   1.55556   1.5625    1.57895   1.59091   1.6087
   1.54545   1.59259   1.60714   1.60784   1.61017
   1.58621   1.59375   1.60606   1.60656   1.60938
   1.59459   1.60526   1.60563   1.60811   1.61039

Mathematica

x = GoldenRatio; numRows = 6; numCols = 12;
usedValues = {}; lowerArray = Table[Null, {numRows}, {numCols}];
For[n = 1, n <= numRows, n++, b = 1;
  foundStart = False;
  While[! foundStart, a = Floor[b*x]; val = a/b;
   If[val < x && GCD[a, b] == 1 && ! MemberQ[usedValues, val],
    lowerArray[[n, 1]] = val;
    AppendTo[usedValues, val]; foundStart = True, b++]];
  For[k = 1, k < numCols, k++, currentVal = lowerArray[[n, k]];
   currentB = Denominator[currentVal];
   nextVal = Null; testB = currentB + 1;
   While[nextVal === Null, testA = Floor[testB*x];
    candidate = testA/testB;
    If[GCD[testA, testB] == 1 && candidate > currentVal &&
      candidate < x && ! MemberQ[usedValues, candidate],
     nextVal = candidate;
     AppendTo[usedValues, candidate];
     lowerArray[[n, k + 1]] = nextVal, testB++]]; ]; ];
Grid[lowerArray]  (* Lower best approximating array to golden ratio *)
Grid[N[lowerArray]]  (* approximations *)
Grid[Denominator[lowerArray]]  (* A394033 *)
Grid[Numerator[lowerArray]]    (* A394034 *)

Crossrefs

Cf. A001622, A001519 (row 1), A394035, A394036, A394037.

Sequence in context: A333573 A361852 A032459 * A263647 A028870 A379680

Adjacent sequences: A394029 A394030 A394031 * A394034 A394035 A394036

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Mar 09 2026

Status

approved