# 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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### 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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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: $$f_n = f_{n-1} + f_{n-2}: ... 1answer 4k views ### Solving a question about Fibonacci and Lucas numbers using induction Im working on practice problems that the instructor gave us yesterday, and I absolutely have no clue of how to solve this problem.. I need to use mathmatical induction to solve this problem.. The ... 2answers 4k views ### Show that Fibonacci and Lucas numbers satisfy the following equality for all n ≥ 2. Fibonacci numbers F1, F2, F3, . . . are defined by the rule: F1 = F2 = 1 and Fk = Fk−2 + Fk−1 for k > 2. Lucas numbers L1, L2, L3, . . . are defined in a similar way by the rule: L1 = 1, L2 = 3 and ... 0answers 490 views ### Relationship of powers of Phi to Lucas Numbers I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ... 1answer 170 views ### What's the Lucas version of the Möbius test for Fibonacci numbers? I recently came across the following, attributed to Möbius:$$(a\in\mathbb N)=F_n\iff\left[\varphi a-\tfrac{1}{a},\varphi a+\tfrac{1}{a}\right]\ni(b\in\mathbb N)$$It is the lesser-known test used to ... 1answer 175 views ### Are some Lucas numbers always coprime with all previous Lucas numbers? I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively ... 1answer 85 views ### What is the Lucas counterpart to the Fibonacci identity 5F_n^2\pm~4=\lambda^2? It's a well-known rule that a number x belongs to the Fibonnaci Sequence iff:$$\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 ... 1answer 80 views ### 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. 1answer 70 views ### 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 ... 1answer 128 views ### Converting Fibonacci number F_{5n+3} to Lucas numbers L_{n+k} I'm trying to prove thatF_{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 ... 1answer 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 ... 1answer 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 ... 1answer 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 ... 3answers 1k views ### 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 ... 1answer 2k views ### Proof about lucas numbers. Define the Lucas numbers to bel_n = l_{n-1} + l_{n-2} $$if n \ge 2 with initial conditions l_0 = 2 and l_1= 1. I "proved" by induction that l_n = f_{n-1} + f_{n+1} for n \ge 1 (by ... 0answers 1k views ### Fibonacci and Lucas numbers related identities We know that H_n = L_n + mF_n, where n = 0 or n > 0 is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ... 4answers 1k views ### Reccurence relation: Lucas sequence I need to solve the given recurrence relation:$$L_n = L_{n-1} + L_{n-2},$$n\geq3 and  L_1 = 1, L_2 =3 I'm confused as to what n\geq3 is doing there, since L_1 and L_2 are given I got ... 0answers 1k views ### Closed formula for Lucas numbers [duplicate] Possible Duplicate: Prove this formula for the Fibonacci Sequence How does one find a formula for the recurrence relation a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}? How do I go about obtaining a ... 1answer 338 views ### An identity involving Lucas numbers Let L_n be the Lucas numbers, defined by L_n = F_{n-1} + F_{n+1} where f is the Fibonacci numbers. How to prove that L_{2n+1} = \displaystyle \sum_{k=0}^{\lfloor n + 1/2\rfloor}\frac{2n+1}{2n+1 ... 2answers 436 views ### Fibonacci/Lucas Number Congruences Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new.$$ ...
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 ...
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 ...