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Let <inline-formula><math 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 Fibonacci sequence given by <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>F</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, for <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>F</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. There are several interesting identities involving this sequence such as <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>F</mi><mi>n</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>F</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>=</mo><msub><mi>F</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula>, for all <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>. In 2012, Chaves, Marques and Togbé proved that if <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> is a linear recurrence sequence (under weak assumptions) and <inline-formula><math display="inline"><semantics><mrow><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mi>s</mi></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow><mi>s</mi></msubsup><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula>, for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ł</i> and the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula>. In this paper, we shall prove that if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>x</mi><mo>ℓ</mo></msub><mo>)</mo></mrow></semantics></math></inline-formula> is an integer homogeneous <i>s</i>-degree polynomial (under weak hypotheses) and if <inline-formula><math display="inline"><semantics><mrow><mi>P</mi><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>G</mi><mrow><mi>n</mi><mo>+</mo><mo>ℓ</mo></mrow></msub><mo>)</mo></mrow><mo>∈</mo><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></mrow></semantics></math></inline-formula> for infinitely many positive integers <i>n</i>, then <i>s</i> is bounded by an effectively computable constant depending only on <i>ℓ</i>, the parameters of <inline-formula><math display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>G</mi><mi>m</mi></msub><mo>)</mo></mrow><mi>m</mi></msub></semantics></math></inline-formula> and the coefficients of <i>P</i>.