A394036 Rectangular array, read by descending antidiagonals: numerators in the best upper approximating array to the golden ratio, phi (A001622). (See A394035.)

2, 5, 7, 13, 12, 9, 34, 17, 22, 28, 89, 18, 23, 51, 33, 233, 31, 41, 59, 64, 38, 610, 44, 49, 67, 75, 77, 43, 1597, 47, 57, 70, 83, 80, 85, 106, 4181, 81, 60, 73, 86, 96, 101, 119, 111, 10946, 115, 107, 133, 159, 99, 109, 122, 127, 145, 28657, 123, 128, 154, 167, 185, 112, 125, 135, 148, 153, 75025, 212, 149, 175, 196, 193, 206, 211, 138, 151, 161, 174

Offset

1,1

Links

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

Example

Corner of array of denominators:
      1   3   8  21  55  144  377  987 2584 6765 17711
      4   7  10  11  19   27   29   50   71   76   131
      5  13  14  25  30   35   37   66   79   92    97
     17  31  36  41  43   45   82   95  108  113   118
     20  39  46  51  53   98  103  121  134  139   207
     23  47  49  59  61  114  119  124  129  155   160
Corner of array of numerators:
      2   5  13  34  89  233  610  1597  4181  10946  28657
      7  12  17  18  31   44   47    81   115    123    212
      9  22  23  41  49   57   60   107   128    149    157
     28  51  59  67  70   73  133   154   175    183    191
     33  64  75  83  86  159  167   196   217    225    335
     38  77  80  96  99  185  193   201   209    251    259
Corner of array of approximations:
     2.       1.66667   1.625    1.61905  1.61818  1.61806
     1.75     1.71429   1.7      1.63636  1.63158  1.62963
     1.8      1.69231   1.64286  1.64     1.63333  1.62857
     1.64706  1.64516   1.63889  1.63415  1.62791  1.62222
     1.65     1.64103   1.63043  1.62745  1.62264  1.62245
     1.65217  1.6383    1.63265  1.62712  1.62295  1.62281

Mathematica

x = GoldenRatio; numRows = 6; numCols = 12;
usedValuesUpper = {}; upperArray = Table[Null, {numRows}, {numCols}];
For[n = 1, n <= numRows, n++, b = 1; foundStart = False;
  While[! foundStart, a = Ceiling[b*x]; val = a/b;
   If[val > x && GCD[a, b] == 1 && ! MemberQ[usedValuesUpper, val],
    upperArray[[n, 1]] = val; AppendTo[usedValuesUpper, val]; foundStart = True, b++]];
  For[k = 1, k < numCols, k++, currentVal = upperArray[[n, k]];
   currentB = Denominator[currentVal]; nextVal = Null; testB = currentB + 1;
   While[nextVal === Null, testA = Ceiling[testB*x];
    candidate = testA/testB; If[GCD[testA, testB] == 1 && candidate < currentVal &&
      candidate > x && ! MemberQ[usedValuesUpper, candidate],
     nextVal = candidate; AppendTo[usedValuesUpper, candidate];
     upperArray[[n, k + 1]] = nextVal, testB++]]; ]; ];
Grid[upperArray] (* Best lower approximating array to golden ratio*)
Grid[N[upperArray]]  (* Approximations *)
Grid[Denominator[upperArray]]  (* A394035 *)
Grid[Numerator[upperArray]]    (* A394036 *)

Crossrefs

Cf. A001622, A001906 (row 1), A394033, A394034, A394035, A394037, A394038.

Sequence in context: A092792 A238336 A083271 * A094285 A121218 A371239

Adjacent sequences: A394033 A394034 A394035 * A394037 A394038 A394039

Keyword

nonn,tabl,frac

Author

Clark Kimberling, Mar 24 2026

Status

approved