Fibonacci Numbers and Their Lucas Coefficients
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908510389796864 |
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| author | Suthar, Tapan |
| author_facet | Suthar, Tapan |
| contents | We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a representation of the nth Fibonacci number as Fn = (1 / (n-1)) times the same sum. We present a detailed proof by mathematical induction and illustrate the identity with a numeric example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00070 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fibonacci Numbers and Their Lucas Coefficients Suthar, Tapan Number Theory 11B39, 11B83 We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a representation of the nth Fibonacci number as Fn = (1 / (n-1)) times the same sum. We present a detailed proof by mathematical induction and illustrate the identity with a numeric example. |
| title | Fibonacci Numbers and Their Lucas Coefficients |
| topic | Number Theory 11B39, 11B83 |
| url | https://arxiv.org/abs/2509.00070 |