Pell and Pell–Lucas Sequences of Fractional Order
Abstract
1. Introduction
2. Preliminaries
3. Fractional Pell Sequence
3.1. The Solution of the Initial Value Problem (7) and (8)
3.2. Closed-Form Expression of Fractional Pell Sequence
3.3. Numerical Scheme for the Fractional Pell Sequence
3.4. Fractional Silver Ratio and Tiling Obtained with Fractional Pell’s Numbers
3.4.1. Fractional Silver Ratio
3.4.2. Fractional Pell’s Tiling
4. Fractional Pell–Lucas Sequence
4.1. Closed Form of the Fractional Pell–Lucas Sequence
4.2. Numerical Scheme for the Fractional Pell–Lucas Sequence
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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# | IO | |||
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 0 1 1.99000000000 4.97505000000 11.92008817890 28.77036010178 69.35116600455 167.20881853340 403.13188359617 971.93524756072 2343.28804106191 5649.51967020207 13620.45119203567 32836.32193044393 79147.45145012428 | 0 1 1.99500000000 4.98751250000 11.96001913091 28.88507358306 69.67521980455 168.10317529800 405.56193312317 978.45500806016 2360.60539480727 5695.14417610542 13739.88521182454 33147.66861641826 79961.75170649997 | 0 1 1.99900000000 4.99750050000 11.99199983283 28.97699771786 69.93498596635 168.82043801696 407.51174632077 983.68898580072 2374.51490469204 5731.81033968845 13835.92258701109 33398.17489180408 80617.47387422606 | 0 1 2 5 12 29 70 169 408 985 2378 5741 13860 33461 80782 |
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Jonnalagadda, J.M.; Danca, M.-F. Pell and Pell–Lucas Sequences of Fractional Order. Fractal Fract. 2025, 9, 416. https://doi.org/10.3390/fractalfract9070416
Jonnalagadda JM, Danca M-F. Pell and Pell–Lucas Sequences of Fractional Order. Fractal and Fractional. 2025; 9(7):416. https://doi.org/10.3390/fractalfract9070416
Chicago/Turabian StyleJonnalagadda, Jagan Mohan, and Marius-F. Danca. 2025. "Pell and Pell–Lucas Sequences of Fractional Order" Fractal and Fractional 9, no. 7: 416. https://doi.org/10.3390/fractalfract9070416
APA StyleJonnalagadda, J. M., & Danca, M.-F. (2025). Pell and Pell–Lucas Sequences of Fractional Order. Fractal and Fractional, 9(7), 416. https://doi.org/10.3390/fractalfract9070416