ClearAll[FindMetallicRatio]
FindMetallicRatio[b_, digits_] :=
 Module[{n, m, data, acc, old, done = False},
  {n, m} = {1, 1};
  old = -100;
  data = {};
  While[done == False,
   {n, m} = {m, b m + n};
   AppendTo[data, {m, m/n}];
   If[Length[data] > 15,
    If[-N[Log10[Abs[data[[-1, 2]] - data[[-2, 2]]]]] > digits,
     done = True
     ]
    ]
   ];
  acc = -N[Log10[Abs[data[[-1, 2]] - data[[-2, 2]]]]];
  <|"sequence" -> Join[{1, 1}, data[[All, 1]]],
   "ratio" -> data[[All, 2]], "acc" -> acc,
   "steps" -> Length[data]|>
  ]
Do[
 out = FindMetallicRatio[b, 32];
 Print["b=", b];
 Print["b=", b, " first 15=", Take[out["sequence"], 15]];
 Print["b=", b, " ratio=", N[Last[out["ratio"]], {\[Infinity], 33}]];
 Print["b=", b, " Number of steps=", out["steps"]];
 ,
 {b, 0, 9}
 ]
out = FindMetallicRatio[1, 256];
out["steps"]
N[out["ratio"][[-1]], 256]