Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences
In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow>&...
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2024-11-01
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| author | Eudes A. Costa Paula M. M. C. Catarino Francival S. Monteiro Vitor M. A. Souza Douglas C. Santos |
| author_facet | Eudes A. Costa Paula M. M. C. Catarino Francival S. Monteiro Vitor M. A. Souza Douglas C. Santos |
| author_sort | Eudes A. Costa |
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| description | In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>{</mo><msub><mi>f</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><msub><mi mathvariant="double-struck">N</mi><mn>0</mn></msub></mrow></msub></semantics></math></inline-formula> having initial terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" 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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and recurrence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>=</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mspace width="4pt"></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mi>n</mi><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula>, with initial terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mn>0</mn><mo>∗</mo></msubsup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>f</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>3</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is generated by the recurrence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mi>n</mi><mo>∗</mo></msubsup><mo>=</mo><mi>t</mi><msubsup><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>∗</mo></msubsup><mo>+</mo><mi>t</mi><msubsup><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>∗</mo></msubsup><mspace width="4pt"></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities. |
| format | Article |
| id | doaj-art-b3caa77b278f41b4ba8001e98462ef44 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-b3caa77b278f41b4ba8001e98462ef442025-08-20T01:55:35ZengMDPI AGMathematics2227-73902024-11-011223372310.3390/math12233723Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type SequencesEudes A. Costa0Paula M. M. C. Catarino1Francival S. Monteiro2Vitor M. A. Souza3Douglas C. Santos4Department of Mathematics, Federal University of Tocantins, Arraias 77330-000, BrazilDepartment of Mathematics, University of Trás-os-Montes and Alto Douro, 5000-801 Vila Real, PortugalDepartment of Mathematics, Federal University of Tocantins, Arraias 77330-000, BrazilDepartment of Mathematics, Federal University of Tocantins, Arraias 77330-000, BrazilEducation Department of the State of Bahia, Barreiras 41745-004, BrazilIn this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>{</mo><msub><mi>f</mi><mi>n</mi></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>∈</mo><msub><mi mathvariant="double-struck">N</mi><mn>0</mn></msub></mrow></msub></semantics></math></inline-formula> having initial terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" 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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> and recurrence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mo>=</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mspace width="4pt"></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mi>n</mi><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula>, with initial terms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mn>0</mn><mo>∗</mo></msubsup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>f</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>=</mo><mrow><mo>(</mo><msubsup><mi>f</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>2</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>f</mi><mn>3</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is generated by the recurrence relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><msubsup><mi>f</mi><mi>n</mi><mo>∗</mo></msubsup><mo>=</mo><mi>t</mi><msubsup><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mo>∗</mo></msubsup><mo>+</mo><mi>t</mi><msubsup><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mo>∗</mo></msubsup><mspace width="4pt"></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities.https://www.mdpi.com/2227-7390/12/23/3723Binet’s formulaFibonacci-type sequencesTricomplex Fibonacci sequencegenerating function |
| spellingShingle | Eudes A. Costa Paula M. M. C. Catarino Francival S. Monteiro Vitor M. A. Souza Douglas C. Santos Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences Mathematics Binet’s formula Fibonacci-type sequences Tricomplex Fibonacci sequence generating function |
| title | Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences |
| title_full | Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences |
| title_fullStr | Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences |
| title_full_unstemmed | Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences |
| title_short | Tricomplex Fibonacci Numbers: A New Family of Fibonacci-Type Sequences |
| title_sort | tricomplex fibonacci numbers a new family of fibonacci type sequences |
| topic | Binet’s formula Fibonacci-type sequences Tricomplex Fibonacci sequence generating function |
| url | https://www.mdpi.com/2227-7390/12/23/3723 |
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