Continued Fractions, a-Fibonacci numbers, and the middle b-noise

Authors

  • Aakash Gurung Juniata University
  • Cheng-Han Pan Juniata College

DOI:

https://doi.org/10.33043/r6a225bc

Abstract

Problem 1186 in The College Mathematics Journal asked for a closed form expression of the continued fraction [1;1; : : : ;1;3;1;1; : : : ;1], and reappeared as Problem 1385 in the PME journal. In this paper, we present a generalization to [a;a; : : : ;a;b;a;a; : : : ;a] with a-Fibonacci numbers and discuss how much the middle b-noise would impact the continued fractions with all a’s.

Author Biographies

Aakash Gurung, Juniata University

Aakash Gurung is a junior at the University of Alabama, pursuing a dual major in Mathematics and Physics. This project was undertaken during his time at Juniata College. He is currently exploring diverse areas of mathematics, driven by a long-term aspiration to pursue a career in mathematical research.

Cheng-Han Pan, Juniata College

Cheng-Han Pan received his Ph.D. in mathematics from West Virginia University. He served as a visiting assistant professor and faculty advisor for Juniata Problem Solving Group at Juniata College before joining Western New England University. His research interests focus on the foundations of real analysis, especially exploring paradoxical examples of functions and sets.

References

J. Sutherland Frame, Continued Fractions and Matrices, The American Mathematical Monthly, Taylor & Francis, Vol. 56, No. 2, 1949, pp. 98–103. DOI: 10.2307/2306169.

Thomas Koshy, Pell and PellLucas Numbers with Applications, 1st ed., Springer New York, NY, 2014, pp. XXIII+431. ISBN: 978-1-4614-8488-2. DOI: 10.1007/978-1-4614-8489-9.

Steven J. Miller, Problem Department, The Pi Mu Epsilon Journal, Vol. 15, No. 6, 2022, pp. 374–375.

Steven J. Miller, Problem Department, The Pi Mu Epsilon Journal, Vol. 15, No. 7, 2022, pp. 445–446.

Greg Oman and Charles N. Curtis, Problems and Solutions, The College Mathematics Journal, Taylor & Francis, Vol. 51, No. 5, 2020, pp. 386–392. DOI: 10.1080/07468342.2020.1826771.

Greg Oman and Charles N. Curtis, Problems and Solutions, The College Mathematics Journal, Taylor & Francis, Vol. 52, No. 5, 2021, pp. 388–396. DOI: 10.1080/07468342.2021.1969181.

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Published

2025-03-28

How to Cite

Gurung, A., & Pan, C.-H. (2025). Continued Fractions, a-Fibonacci numbers, and the middle b-noise. Mathematics Exchange, 18(1), 77–87. https://doi.org/10.33043/r6a225bc

Issue

Vol. 18 No. 1 (2024): Mathematics Exchange

Section

Articles

License

Copyright (c) 2024 Aakash Gurung, Cheng-Han Pan

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.