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Fibonacci Numbers and Their Lucas Coefficients
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.
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On Concatenations of Fibonacci and Lucas Numbers
Bulletin of the Iranian Mathematical Society, 2022Let \( (F_n)_{n\ge 0} \) and \( (L_n)_{n\ge 0} \) be the usual Fibonacci and Lucas sequences defined respectively by the linear recurrence relations: \( F_0=0 \), \( F_1=1 \), \( F_{n+2}=F_{n+1}+F_n \) and \( L_0=2 \), \( L_1=1 \), \( L_{n+2}=L_{n+1}+L_n \) for all \( n\ge 0 \).
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2021
In the literature, the Fibonacci numbers are usually denoted by \(F_n\), but this symbol is already reserved for the Fermat numbers in this book. So we will denote them by \(K_n\). The sequence of Fibonacci numbers \(\,\,(K_n)_{n=0}^\infty \,\,\) starts with \(K_0=0\) and \(K_1=1\) and satisfies the recurrence.
Michal Křížek +2 more
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In the literature, the Fibonacci numbers are usually denoted by \(F_n\), but this symbol is already reserved for the Fermat numbers in this book. So we will denote them by \(K_n\). The sequence of Fibonacci numbers \(\,\,(K_n)_{n=0}^\infty \,\,\) starts with \(K_0=0\) and \(K_1=1\) and satisfies the recurrence.
Michal Křížek +2 more
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Power sums of Fibonacci and Lucas numbers
Quaestiones Mathematicae, 2011Polynomial representation formulae for power sums of the extended Fibonacci-Lucas numbers are established, which include, as special cases, four for-mulae for odd power sums of Melham type on Fibonacci and Lucas numbers, obtained recently by Ozeki and Prodinger (2009).Quaestiones Mathematicae 34(2011), 75 ...
Chu, Wenchang, Li, Nadia N
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Perfect fibonacci and lucas numbers
Rendiconti del Circolo Matematico di Palermo, 2000Using elementary means, the author shows that no Fibonacci or Lucas number is perfect.
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On matrices related with Fibonacci and Lucas numbers
Applied Mathematics and Computation, 2008In this paper, we obtain some new results on matrices related with Fibonacci numbers and Lucas numbers. Also, we derive the relation between Pell numbers and its companion sequence by using our representations.
Xudan Fu, Xia Zhou
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1997
Consider the following number trick–try it out on your friends. You ask them to write down the numbers from 0 to 9. Against 0 and 1 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers).
Peter Hilton +2 more
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Consider the following number trick–try it out on your friends. You ask them to write down the numbers from 0 to 9. Against 0 and 1 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers).
Peter Hilton +2 more
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On Fibonacci search method with k-Lucas numbers
Applied Mathematics and Computation, 2003zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Yildiz, B, Karaduman, E
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On the representation of k-generalized Fibonacci and Lucas numbers
Applied Mathematics and Computation, 2005zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Ahmet Ali Öcal +2 more
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INTRA RATIOS OF FIBONACCI AND LUCAS NUMBERS
JP Journal of Algebra, Number Theory and Applications, 2015Summary: We study the ratios of any \(k\) spacing apart Fibonacci numbers and Lucas numbers as well as intra ratios \(L_n/F_{n\pm k}\) by means of semisimple continued fraction. And the semisimple continued fractions will be applied to solve certain systems of linear equation.
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