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 $L_1=1$.

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Proofing that the Lucas numbers come closer to the Phi rounded numbers then the Fibonacci numbers.

Morning everyone, Bit of background, I'm a mid level programmer with very limited mathematics skills. As part of an assessment for a new role I've been asked to complete a technical task which ...
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611 views

Proving the closed form of a generating function of the sum of n lucas numbers is equal to the n+2th lucas number

1760887     I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$ $M_n = 1 + \sum_{i=0}^n{...
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117 views

Show that $\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10}$

Show that$$\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10},$$ where $F_n$ is a Fibonacci number and $L_n$ is a Lucas number.$^1$ Motivation: For example, when calculating ...
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Proofs with Fibonacci and Lucas numbers via induction

How would I go about proving the following sequence using induction on $k$? $2F_{2n+k} = F_{n+k}L_n + F_nL_{n+k}$ I know I have to show that it's true for $k = 1$, but I can't even seem to be able ...
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157 views

Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$

Let $\alpha =\left(\frac{1+\sqrt{5}}{2}\right)$ and $\beta = \left(\frac{1-\sqrt{5}}{2}\right)$. Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$ where $L_n$ denotes the Lucas numbers. ...
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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/...
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172 views

Are any factors of Lucas numbers divisible by a Fibonacci number greater than three?

The congruence relation for Fibonacci and Lucas numbers is stated: If Fn > 3 is a Fibonacci number then no Lucas number is divisible by Fn. However, does this apply to the factors as well?
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Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any ...
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Perfect Squares on Lucas Sequences

Let the $f(x) = x^2 -ax+b$ has a positive discriminant $D=a^2-4b$ and $k,l$ be its roots. Then $U_n = \frac{k^n-l^n}{k-l}$ and $V_n=k^n+l^n$. I would like to prove these 4 properties If $U_n$ is a ...
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316 views

Lucas Number Sequence

Can anyone help me in this question: Define $ (b_n)$ as $b_1= 1,b_n=a_{n+1} - a_n $ for $ n\ge 2$, where $ a_n $ is the Fibonnaci series. This sequence is known as the sequence of Lucas numbers. ...
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How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
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143 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = 1,2,\cdots,...
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Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Let $$ \text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction} $$ So for example, the terms of the sum $\text{S}_6$ ...
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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}: ...
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4k views

Solving a question about Fibonacci and Lucas numbers using induction

Im working on practice problems that the instructor gave us yesterday, and I absolutely have no clue of how to solve this problem.. I need to use mathmatical induction to solve this problem.. The ...
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Show that Fibonacci and Lucas numbers satisfy the following equality for all n ≥ 2.

Fibonacci numbers F1, F2, F3, . . . are defined by the rule: F1 = F2 = 1 and Fk = Fk−2 + Fk−1 for k > 2. Lucas numbers L1, L2, L3, . . . are defined in a similar way by the rule: L1 = 1, L2 = 3 and ...
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Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
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What's the Lucas version of the Möbius test for Fibonacci numbers?

I recently came across the following, attributed to Möbius: $$(a\in\mathbb N)=F_n\iff\left[\varphi a-\tfrac{1}{a},\varphi a+\tfrac{1}{a}\right]\ni(b\in\mathbb N)$$ It is the lesser-known test used to ...
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175 views

Are some Lucas numbers always coprime with all previous Lucas numbers?

I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively ...
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What is the Lucas counterpart to the Fibonacci identity $5F_n^2\pm~4=\lambda^2$?

It's a well-known rule that a number $x$ belongs to the Fibonnaci Sequence iff: $$\begin{align}5x^2\pm~4&=\lambda^2&\lambda\in\mathbb Z\end{align}$$ In other words, if and only if $5x^2\pm~4$ ...
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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.
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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 ...
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128 views

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 ...
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128 views

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 ...
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268 views

Prove relation between Lucas and Fibonacci numbers using tilings

I struggle a lot with combinatorial proofs and was hoping for some help. I need to prove by strong induction that $L_n = F_{n-2} + F_n$ and how this shows that $L_n$ counts the tilings of the circular ...
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75 views

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 ...
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Lucas Numbers and Matrices

Is there a $2 \times 2$ matrix that can be raised to any power to obtain the Lucas Numbers? If so, what is that matrix? I've looked around on this website and other sites and am not able to find the ...
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Proof about lucas numbers.

Define the Lucas numbers to be $$l_n = l_{n-1} + l_{n-2} $$ if $n \ge 2$ with initial conditions $l_0 = 2$ and $l_1= 1$. I "proved" by induction that $l_n = f_{n-1} + f_{n+1}$ for $n \ge 1$ (by $...
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Fibonacci and Lucas numbers related identities

We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ...
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Reccurence relation: Lucas sequence

I need to solve the given recurrence relation:$$L_n = L_{n-1} + L_{n-2},$$ $n\geq3$ and $ L_1 = 1, L_2 =3$ I'm confused as to what $n\geq3$ is doing there, since $L_1$ and $L_2$ are given I got $...
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Closed formula for Lucas numbers [duplicate]

Possible Duplicate: Prove this formula for the Fibonacci Sequence How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$ How do I go about obtaining a ...
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An identity involving Lucas numbers

Let $L_n$ be the Lucas numbers, defined by $L_n = F_{n-1} + F_{n+1}$ where f is the Fibonacci numbers. How to prove that $L_{2n+1} = \displaystyle \sum_{k=0}^{\lfloor n + 1/2\rfloor}\frac{2n+1}{2n+1 ...
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Fibonacci/Lucas Number Congruences

Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new. $$ ...
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Common terms in general Fibonacci sequences

Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
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Lucas Numbers and Tilings

Show that $f_{n-1} + L_n = 2f_{n}$. So we need to find a $2$ to $1$ correspondence. Set 1: Tilings an $n$-board. Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet. So we need to ...

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