Questions tagged [lucas-numbers]

Questions on the Lucas numbers, a special sequence of integers that satisfy the recurrence $L_n=L_{n-1}+L_{n-2}$ with the initial conditions $L_0=2$ and $L_1=1$.

The $n$th Lucas number $L_n$ is defined recursively, by

$$L_n = L_{n - 1} + L_{n - 2}$$

for $n > 1$, and $L_0 = 2,\; L_1 = 1$. There is a closed form expression, namely

$$ L_{n}=\varphi ^{n}+(1-\varphi )^{n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n},$$

where the golden ratio $\varphi$ is equal to $\dfrac{1 + \sqrt{5}}{2}$.

The sequence of Lucas numbers is: $2,1,3,4,7,11,18,29,47,76,123,\dots$ (sequence A000032 in the OEIS).

Reference: Lucas number.

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How to prove this Fibonacci identity? $\sum_{k=0}^{n} F_{k} F_{n-k} = \frac{1}{5}\left(n L_{n} - F_{n}\right)$

How to prove this Fibonacci identity? $$\sum_{k=0}^{n-3} F_{k} F_{n-k-3} = \frac{(n-3)L_{n-3} - F_{n-3}}{5}$$ i tried to used the generating function and partial decomposition but i got confused?
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Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points of Fibonacci factors. It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://webspace.ship.edu/msrenault/fibonacci/FibThesis.pdf If m has prime…
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Fibonacci and Lucas series technique

Well, I have the following two problems involving Fibonacci sequences and Lucas numbers. I know that they share the same technique, but I don't have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 =0, f_1=1$$ $$l_n=l_{n-1}…
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Lucas Numbers Inequality

Can it be shown that \begin{align} \frac{1}{\ln(1+L_{n}) -1} \geq \frac{L_{n}}{(L_{n}-1)(e^{L_{n}}-1)} \end{align} where $L_{n}$ is the $n^{th}$ Lucas number. Show results in full detail.
Leucippus
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Prove that for $n\ge 2$, the n-th Lucas number is equal to $[a^n+1/2]$

Prove that for n greater than or equal to 2 the n-th Lucas number is equal to $[a^n+1/2]$. The brackets are the greatest integer function, $a = \frac{1+\sqrt5}{2}$. I get every kind of proof we have done in number theory except the greatest…
user142903
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Converting Fibonacci number $F_{5n+3}$ to Lucas numbers $L_{n+k}$

I'm trying to prove that$F_{5n+3}\text{mod}10 = L_{n}\text{mod}10$. I rearranged it into a more solvable form of $F_{5n+3}-L_n = 10k$ (because if two numbers end in the same digit, their difference must be 10 and vice versa). The problem is when I…
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Proof Help dealing Lucas and Fibonacci Numbers

Claim: $L_n=F_{n-1}+F_{n+1}$ for all $n >0$ Could someone please help me prove this? My professor mentioned it in class, but didn't show us how to prove it. I am just curious. The $L$ stands for the Lucas numbers and the $F$ stands for the Fibonacci…
A Glenn
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Closed form representation for $\sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}k$

Answering some other question, I stumbled upon the following relationship: For $n\in\Bbb N$ let $$p_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}k$$ and let $$a_n = p_n+p_{n-2}\quad \text{ if } n \geqslant 2; \qquad (a_1,a_2) = (1,3)$$ then $a_n…
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How to prove that $a_{2n} = a_nb_n$ in a Lucas sequence?

Here's the question in in my book: Define $(b_n)$ by $b_1=1$, $b_n = a_{n+1}+a_{n-1}$ for $n ≥ 2$. $(b_n)$ is known as the sequence of Lucas numbers. Prove (i) $b_n = b_{n-1} + b_{n-2}$ for $n ≥ 3$. (ii) $a_{2n} = a_nb_n$. where $(a_n)$ is the…
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Primality test for numbers of the form $(10^p-1)/9$ (and maybe $((10 \cdot 2^n)^p-1)/(10 \cdot 2^n-1)$)

This question is successor of Primality test for numbers of the form (11^p−1)/10 Here is what I observed: For $(10^p-1)/9$ : Let $N$ = $(10^p-1)/9$ when $p$ is a prime number $p > 3$. Let the sequence $S_i=S_{i-1}^{10}-10 S_{i-1}^8+35 S_{i-1}^6-50…
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A Lucas sequence

Define a recursive sequence $\{x_n\}$ as follows $$x_0=2,x_1=3,x_n=\frac{x_{n-1}^2+5}{x_{n-2}}(n\ge2).$$ Prove $x_n$ is a prime if and only if $n=0$ or $n=2^k$ where $k\in \mathbb{N}.$ Note that \begin{align*}…
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How can I verify the ratio of Lucas numbers to fibonacci numbers algebreically

The article: "The Lucas numbers 1,3,4...are the sums of alternate Fibonacci numbers. The ratios of Lucas to Fibonacci must satisfy: $R_j = \frac{F_{i+1}+F_{i-1}}{F_i}=\frac{2F_{i+1}}{F_i-1}$ I know that Fibonacci numbers and Lucas are the same…
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if $p\mid u_m$, $m\mid n$, $p\mid u_n/u_m$, prove that $p\mid n/m$

If we have that $p\mid u_a$, $b\mid a$, and $p\mid u_a/u_b$, prove that $p\mid n/b$, assuming that $u_a$ and $u_b$ are terms in the linear recurrence for the Lucas Sequence. I've tried looking at the characteristic polynomial and using the…
analie
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Lowest bounds of Lucas Numbers

I'm currently working with bounding terms of a recurrence relation and just filled out the table for $L_n < (1.7)^n$ and am asked to figure out why the number $1.7$ is so special and how I can adjust what I've already done to create the tightest…
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How to show that : $4(-1)^nL_n^2+L_{4n}-L_n^4=2$

How can we prove that: $$4(-1)^nL_n^2+L_{4n}-L_n^4=2$$ Where $L_n$ is Lucas number We got…
Sibawayh
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