A394034 Rectangular array, read by descending antidiagonals: numerators in the best lower approximating array to the golden ratio, phi (A001622). (See A394033.)

1, 3, 4, 8, 11, 14, 21, 19, 25, 17, 55, 27, 30, 43, 46, 144, 29, 35, 45, 51, 59, 377, 50, 37, 82, 53, 61, 67, 987, 71, 66, 95, 98, 114, 69, 77, 2584, 76, 79, 108, 103, 119, 127, 140, 85, 6765, 131, 92, 113, 121, 124, 137, 153, 156, 93, 17711, 186, 97, 118, 134

Offset

1,2

Links

Table of n, a(n) for n=1..60.

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: A287404 A287388 A022888 * A287454 A287446 A051207

Adjacent sequences: A394030 A394031 A394033 * A394035 A394036 A394037

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Mar 10 2026

Status

approved