| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>F</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> be the sequence of Fibonacci numbers. The order of appearance of an integer <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> is defined as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mo movablelimits="true" form="prefix">min</mo><mrow><mo>{</mo><mi>k</mi><mo>≥</mo><mn>1</mn><mo>:</mo><mi>n</mi><mo>∣</mo><msub><mi>F</mi><mi>k</mi></msub><mo>}</mo></mrow></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> be the set of all limit points of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mi>z</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>/</mo><mi>n</mi><mo>:</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula>. By some theoretical results on the growth of the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><mi>z</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>/</mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula>, we gain a better understanding of the topological structure of the derived set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. For instance, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mn>2</mn><mo>}</mo></mrow><mo>⊆</mo><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>⊆</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula> does not have any interior points. A recent result of Trojovská implies the existence of a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup><mo>∩</mo><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is the empty set. In this paper, we improve this result by proving that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mfrac><mn>12</mn><mn>7</mn></mfrac><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> is the largest subinterval of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula> which does not intersect <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">Z</mi><mo>′</mo></msup></semantics></math></inline-formula>. In addition, we show a connection between the sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mi>n</mi></msub></semantics></math></inline-formula>, for which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>z</mi><mrow><mo>(</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>/</mo><msub><mi>x</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> tends to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> (as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>), and the number of preimages of <i>r</i> under the map <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>↦</mo><mi>z</mi><mo>(</mo><mi>m</mi><mo>)</mo><mo>/</mo><mi>m</mi></mrow></semantics></math></inline-formula>.
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