Using properties of the generalised Zeckendorf decompositions, we study the extended Frobenius problem for sequences of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_513_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\{p_a+p_n\}_{n\in {\mathbb {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mi>a</mi> </msub> <mo>+</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_513_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$\{p_n\}_{n\in {\mathbb {N}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is an r-Fibonacci sequence and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40065_2025_513_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$p_a$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>a</mi> </msub> </math></EquationSource> </InlineEquation> is an r-Fibonacci number.

The extended Frobenius problem for r-Fibonacci sequences shifted by r-Fibonacci numbers

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2025年5月18日

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Juan Manuel López-Medel,
Aureliano M. Robles-Pérez,
José Carlos Rosales

摘要

Using properties of the generalised Zeckendorf decompositions, we study the extended Frobenius problem for sequences of the form $\{p_a+p_n\}_{n\in {\mathbb {N}}}$ ${p_{a} + p_{n}}_{n \in N}$ , where $\{p_n\}_{n\in {\mathbb {N}}}$ ${p_{n}}_{n \in N}$ is an r-Fibonacci sequence and $p_a$ $p_{a}$ is an r-Fibonacci number.