<p>We consider second-order generalized Lucas sequences <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(V_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$V_{n+2} = rV_{n+1} + sV_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>r</mi> <msub> <mi>V</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>s</mi> <msub> <mi>V</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$V_0 = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$V_1 = r$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>r</mi> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$r \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$s \in \{-1, 1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. For the fifteen families whose dominant root satisfies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha < 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo><</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>, we determine all terms that are concatenations of two repdigits in base ten, finding exactly nine solutions. This complements the work of Bravo et al. (Result Math. 76(3):139, 2021) on <i>k</i>-generalized Lucas sequences.</p>

On concatenations of two repdigits in second-order Lucas sequences

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发表日期：

2026年3月18日

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Asmae El-Baz

摘要

We consider second-order generalized Lucas sequences $(V_n)$ $(V_{n})$ defined by $V_{n+2} = rV_{n+1} + sV_n$ $V_{n + 2} = r V_{n + 1} + s V_{n}$ with $V_0 = 2$ $V_{0} = 2$ , $V_1 = r$ $V_{1} = r$ , where $r \ge 1$ $r \geq 1$ and $s \in \{-1, 1\}$ $s \in {- 1, 1}$ . For the fifteen families whose dominant root satisfies $\alpha < 10$ $α < 10$ , we determine all terms that are concatenations of two repdigits in base ten, finding exactly nine solutions. This complements the work of Bravo et al. (Result Math. 76(3):139, 2021) on k-generalized Lucas sequences.