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.

98 questions
0
votes
1 answer

Prove that $\gcd(F_{3K},L_{3k}) \equiv 2$

Following this definition $L_K = F_{K-1} + F_{K+1}$ We have that $\gcd(F_{3K},L_{3k}) = \gcd(F_{3k}, F_{3k+1} + F_{3k-1}) =\gcd(F_{3k}, 2F_{3k-1})$ I don't know where to go from here. How do I prove that $\gcd(F_{3k}, 2F_{3k-1})\equiv 2$
0
votes
1 answer

Lucas Number Questions!

Problem: Find $(a,b)$ such that $$L_n = a\phi^n + b\widehat{\phi}^n.$$ Where $n$ is the $n^{th}$ lucas number. How would I start this? Would I just start by plugging in $a=b=1$ and then trying to solve?
JenkinsMa
  • 405
  • 3
  • 9
0
votes
2 answers

Why does the fibonacci series start with 0 and the lucas series with 1?

Why the difference? And when we're deriving these series from eigenvectors, what difference does the starting point make? Please help. I'm very confused. I have a test tomorrow and need to know the answer to this question. There's literally nothing…
0
votes
0 answers

Finding a closed formula for the nth Lucas Number

The Lucas numbers are defined by $$L_0 = 1, L_1=3$$ $$L_n = L_{n-2} + L_{n-1}$$ I used this knowledge to get an equation for the nth Lucas number as follows: $$L(x) = \frac{1+2x}{1-x-x^2}$$ Now I am asked to find a closed formula for this…
0
votes
1 answer

Is there a proof for why the difference between the n-th power of phi and the n-th Lucas number converges to zero?

Let $\epsilon(n)$ be the absolute value of the difference between the $n$th Lucas number ($L(n)$) and the $n$th power of $\phi$. $\epsilon(n)$ pretty clearly converges to zero, and does so pretty fast. But is there a proof for $$ \lim_{n \to \infty}…
Chris vCB
  • 507
  • 4
  • 7
0
votes
1 answer

Lucas Numbers Matrix A

In linear algebra I have an equation $x_n = Ax_0$. I know the values of $x$ for any given value of $n$, and I know $x_0$. Both are $2\times 1$ matrices. How do I solve for $A$? The answer should be a $2\times2$ matrix with entries $[1 1; 1 0]$. …
0
votes
0 answers

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 mirrors the title of this question. The programming…
0
votes
1 answer

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 $L_1 = 1$ $M_n = 1 + \sum_{i=0}^n{L_i}$ Calculate the first few elements of the…
1 2 3 4 5 6
7