A non negative integer is curious if when written in base ten it is a repdigit one or it is one of the form <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq1.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$ \underbrace{b\cdots b}_{\ell ~\text {times}}\underbrace{a\cdots a}_{m~\text {times}}\underbrace{b\cdots b}_{\ell ~\text {times}} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <munder accentunder="true"> <mrow> <mi>b</mi> <mo>⋯</mo> <mi>b</mi> </mrow> <mo>⏟</mo> </munder> <mrow> <mi>ℓ</mi> <mspace width="3.33333pt" /> <mtext>times</mtext> </mrow> </msub> <msub> <munder accentunder="true"> <mrow> <mi>a</mi> <mo>⋯</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> <mrow> <mi>m</mi> <mspace width="3.33333pt" /> <mtext>times</mtext> </mrow> </msub> <msub> <munder accentunder="true"> <mrow> <mi>b</mi> <mo>⋯</mo> <mi>b</mi> </mrow> <mo>⏟</mo> </munder> <mrow> <mi>ℓ</mi> <mspace width="3.33333pt" /> <mtext>times</mtext> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$\ell ,m\geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>,</mo> <mi>m</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$0\leqslant a\leqslant 9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>⩽</mo> <mi>a</mi> <mo>⩽</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$1\leqslant b\leqslant 9$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>⩽</mo> <mi>b</mi> <mo>⩽</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$a\ne b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>. Now, let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_n)_{n\geqslant 0}$</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> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> be the Padovan sequence given by <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="143" /> </InlineMediaObject> <EquationSource Format="TEX">$P_0=0, P_1=P_2=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>P</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and the recurrence formula <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="129" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{n+3}=P_{n+1}+P_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_794_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \geqslant 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In this note we calculate all Padovan numbers which are curious.

Curious Padovan numbers

期刊论文
发表日期：

2025年5月14日

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Ana Cecilia García Lomelí,
Santos Hernández Hernández

摘要

A non negative integer is curious if when written in base ten it is a repdigit one or it is one of the form $ \underbrace{b\cdots b}_{\ell ~\text {times}}\underbrace{a\cdots a}_{m~\text {times}}\underbrace{b\cdots b}_{\ell ~\text {times}} $ ${\underset{⏟}{b \dots b}}_{ℓ times} {\underset{⏟}{a \dots a}}_{m times} {\underset{⏟}{b \dots b}}_{ℓ times}$ where $\ell ,m\geqslant 1$ $ℓ, m ⩾ 1$ , $0\leqslant a\leqslant 9$ $0 ⩽ a ⩽ 9$ , $1\leqslant b\leqslant 9$ $1 ⩽ b ⩽ 9$ and $a\ne b$ $a \neq b$ . Now, let $(P_n)_{n\geqslant 0}$ ${(P_{n})}_{n ⩾ 0}$ be the Padovan sequence given by $P_0=0, P_1=P_2=1$ $P_{0} = 0, P_{1} = P_{2} = 1$ and the recurrence formula $P_{n+3}=P_{n+1}+P_n$ $P_{n + 3} = P_{n + 1} + P_{n}$ for all $n \geqslant 0$ $n ⩾ 0$ . In this note we calculate all Padovan numbers which are curious.