<i>p</i>-Numerical Semigroups of Triples from the Three-Term Recurrence Relations

<i>p</i>-Numerical Semigroups of Triples from the Three-Term Recurrence Relations

Many people, including Horadam, have studied the numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-...

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Bibliographic Details
Main Authors:	Jiaxin Mu, Takao Komatsu
Format:	Article
Language:	English
Published:	MDPI AG 2024-09-01
Series:	Axioms
Subjects:	Frobenius problem Frobenius numbers Horadam numbers Apéry set recurrence
Online Access:	https://www.mdpi.com/2075-1680/13/9/608
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Summary:	Many people, including Horadam, have studied the numbers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>W</mi><mi>n</mi></msub></semantics></math></inline-formula>, satisfying the recurrence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mi>n</mi></msub><mo>=</mo><mi>u</mi><msub><mi>W</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>v</mi><msub><mi>W</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>) with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. In this paper, we study the <i>p</i>-numerical semigroups of the triple <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mi>W</mi><mi>i</mi></msub><mo>,</mo><msub><mi>W</mi><mrow><mi>i</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>W</mi><mrow><mi>i</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>)</mo></mrow></semantics></math></inline-formula> for integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>,</mo><mi>k</mi><mo>(</mo><mo>≥</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. For a nonnegative integer <i>p</i>, the <i>p</i>-numerical semigroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>S</mi><mi>p</mi></msub></semantics></math></inline-formula> is defined as the set of integers whose nonnegative integral linear combinations of given positive integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>κ</mi></msub></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>gcd</mi><mo>(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>a</mi><mi>κ</mi></msub><mo>)</mo><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> are expressed in more than <i>p</i> ways. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>=</mo><msub><mi>S</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> is the original numerical semigroup. The largest element and the cardinality of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">N</mi><mn>0</mn></msub><mrow><mo>∖</mo></mrow><msub><mi>S</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> are called the <i>p</i>-Frobenius number and the <i>p</i>-genus, respectively.
ISSN:	2075-1680