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For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> integers, let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msubsup><mi>t</mi><mi>n</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> be the sequence of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></semantics></math></inline-formula>-generalized Fibonacci numbers which is defined by the recurrence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>t</mi><mi>n</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mo>=</mo><msubsup><mi>t</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>t</mi><mrow><mi>n</mi><mo>−</mo><mi>r</mi></mrow><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>></mo><mi>r</mi></mrow></semantics></math></inline-formula>, with initial values <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>t</mi><mi>i</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>t</mi><mi>r</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mi>a</mi></mrow></semantics></math></inline-formula>. In this paper, we shall prove (in particular) that, for any given <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>, there exists a positive proportion of positive integers which can not be written as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>t</mi><mi>n</mi><mrow><mo>(</mo><mi>r</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow></msubsup></semantics></math></inline-formula> for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>a</mi><mo>)</mo></mrow><mo>∈</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mi>r</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mrow><mo>≥</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>.