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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$5F_{n+1} = L_{n+4} − L_n.$

I'm very new to induction proof and need some help to show that for $n ∈ N$ we have the relation between the Fibonacci and Lucas numbers: $$5F_{n+1} = L_{n+4} − L_n.$$ I know that I should show true for n = 1 and k = n+1. I also know that the…
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Lucas number identity

Let $L_n$ be the Lucas numbers, defined by the recursion $L_n=L_{n-1}+L_{n-2}$ with initial values $L_0=2$ and $L_1=1$. Any idea how to prove the identity…
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Show that $F_{3n} = F_{n}(L_{2n} + (-1)^n)$

Let $F_n, L_n$ be the Fibonacci and Lucas sequences respectively. Show that $F_{3n} = F_{n}(L_{2n} + (-1)^n)$. In my attempt I am using Binet's formula, and the equivalent for the Lucas numbers. \begin{align} F_{3n} &= \frac {\alpha^{3n} -…
IntegrateThis
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Proof of a Well-Known Fibonacci Identity Involving Cubes of Fibonacci Numbers

The following is due to Lucas in 1876: $$F_{n + 1}^3 + F_n^3 - F_{n - 1}^3 = F_{3n}$$ I am unable to locate an elementary proof of this identity, and am unable to reproduce it myself. Would anyone mind sharing a proof or a source?
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How to construct spiral phyllotactic pattern with the given number of spirals?

It is known that the spiral phyllotactic pattern is common in Nature, especially in Botany. It consists of two group of clockwise and anticlockwise spirals, starting from the center. In most cases the number of those spirals are two consecutive…
Ivan Bunin
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Lucas Reciprocity Laws

Suppose $p$, $q$ are primes such that $p=qk+1$. If $a$ is not $0$, $1,$ or $-1$, then $a^q\equiv1\pmod p$ if and only if $a$ is a $k$-th power residue modulo $p$, so that $a^{p-1}\equiv1\pmod p$. Similarly, for the Fibonacci and Lucas sequences…
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Lucas Sequence and primality tests. is this test deterministic?

consider lucas parameters $(P, Q)$ and $D = P^2 - 4Q$. Let $n>0$,$\big(\frac{D}{n}\big)= - 1$ then $U_{n + 1}\equiv{0 \pmod{n}}$ and $n$ is a Lucas probable prime. This test base only on the divisibility property of second order recurrence relation…
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Pell-Lucas number

I'm studying about Pell number and Pell-Lucas number whose have Binet formula $P_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $Q_n=\alpha^n+\beta^n$, where $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$, respectively. The simple relation between two of…
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Gaussian primes and Lucas numbers

Let $L_{n}$ be the $n$ th Lucas number. For example, $L_{1} = 1, L_{2} = 3, L_{3} = 4$. Conjecture: there is no Gaussian primes in the sequence $(L_{n-1} + L_{n} i)$ for $n = 2$ to $\infty$. I hope that this will be proved or disproved.
SeiichiKirikami
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Fibonacci and Lucas linear relation proofs

I really need some help in doing this: By using the generating functions $F(z)$ and $L(z)$ for Fibonacci and Lucas numbers, show that: $$ F_n = \frac{L_{n-1}}{2}+\frac{L_{n-2}}{2^2}+\ldots+\frac{L_0}{2^n}.$$ I have found that…
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lucas numbers Prove that $l_0^2+l_1^2+...+l_n^2=l_n*l_n+1+2$ for $n \ge 0$

Suppose that the lucas numbers are $l_n=l_{n-1}+l_{n-2}$ for all $n \ge 1$ where $l_0=2$ and $l_1=1$. Prove that $l_0^2+l_1^2+...+l_n^2=l_n*l_{n+1}+2$ for $n \ge 0$ I think the easiest way to prove this would be induction. For the base case when…
AndroidFish
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Why are the Lucas numbers and Fibonacci numbers linearly independent?

The answer for this question states (without giving too much else away): Since $F$ and $L$ are linearly independent ... Using the definition of linear dependence for infinite dimensions, I presume that if I am able to find a finite subset of $L$…
Dair
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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 to do that.
Matt
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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 $t = \frac{1\pm\sqrt 5}{2}$ Which got me the…
pauliwago
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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. I managed to solve the base case: $$L_2 =…
Nick Powers
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