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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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Lucas numbers relation to Φ
So, the Lucas numbers are 2,1,3,4,7,11... Let L(n) be nth lucas number Fibonacci numbers are 1,1,2,3,5,8,13,21... Φ^n=F(n+1)+F(n-1), F=Fibonacci number and n=nth So, if I say n=5, then Φ^n=F(6)+F(4)=... sequences-and-series fibonacci-numbers lucas-numbers- 1
Generalizing Fibonacci/Lucas-like series' in terms of the Fibonacci Sequence
(TLDR: I independently discovered a property of Fibonacci-like sequences and I couldn't have been the first one to notice. Who else has?) For some background, I haven't been in a math class since high-... sequences-and-series fibonacci-numbers lucas-numbers- 11
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 $$a_n = p_n+p_{n-2}\quad \text{ if } n \... sequences-and-series recurrence-relations binomial-coefficients closed-form lucas-numbers- 5,164
How to prove that $a_{2n} = a_nb_n$ in a Lucas sequence? [duplicate]
Here's the question in in my book: Define $(b_n)$ by $b_1=1$, $b_n = a_{n+1}+a_{n-1}$ for $n ≥ 2$. $(b_n)$ is known as the sequence of Lucas numbers. Prove (i) $b_n = b_{n-1} + b_{n-2}$ for $n ≥ 3$. (... elementary-number-theory induction fibonacci-numbers lucas-numbers- 13
How to find the count of edge covers of a graph of degree 2?
I have been trying to understand this editorial for question F of Atcoder beginner contest 247: . What I have not been able to understand is this part:... combinatorics graph-theory algorithms lucas-numbers- 1
A pattern of periodic continued fraction
I am interested in the continued fractions which $1$s are consecutive appears. For example, it is the following values. $$ \sqrt{7} = [2;\overline{1,1,1,4}] \\ \sqrt{13} = [3;\overline{1,1,1,1,6}] $$ ... sequences-and-series number-theory fibonacci-numbers continued-fractions lucas-numbers- 416
Convert LUC formula to GF2 polynomial base
is it possible to convert LUC public key encryption formula from standard math to finite field GF(2) polynomial like RSA? LUC is based on Lucas function and it uses the following equation V(i+1)=mv(i)-... polynomials finite-fields lucas-numbers- 1
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 ... prime-numbers examples-counterexamples primality-test lucas-numbers lucas-lehmer-test- 313
If $(3p_n\pm \sqrt{5p_n^2-4})\Delta/(2p_n)$ is an integer then $p_n$ is an odd-indexed Fibonacci number
Suppose $1<p_1<\dots<p_n$ are pairwise relatively prime integers and let $\Delta=p_1\cdots p_n$. In the proof of Lemma 4.8 of this paper (), there is the ... number-theory elementary-number-theory fibonacci-numbers lucas-numbers- 2,130
Proof recursion is a subset of Lucas numbers
I need to prove that the recursion $a_n=\frac{a_{n-1}^2+5}{a_{n-2}}$ for $a_0=2,a_1=3$ are the Lucas numbers with even index. I would like to use induction, but I got a fraction that I'm not sure how ... induction recurrence-relations lucas-numbers- 39
How can this relation between Lucas and Fibonacci numbers be proved? [closed]
$ \lim_{n \to \infty}\Bigg(\dfrac{2 \cdot 10^n + 1 \cdot 10^{n-1} + 3 \cdot 10^{n-2} + 4 \cdot 10^{n-3}+...}{0\cdot 10^n + 1 \cdot 10^{n-1} + 1 \cdot 10^{n-2} + 2 \cdot 10^{n-3}+...}\Bigg) = 19 $ I ... limits fibonacci-numbers lucas-numbers- 27
How to find a complementary subspace for the subspace of sequences where $u_{n+2}=u_{n} + u_{n+1}$, in $\mathbb{R}^\infty$?
Consider this subspace of $\mathbb{R}^\infty$ (sequences of real numbers): $U = \{\vec{u} = (u_1, u_2, ...) \in \mathbb{R}^\infty | u_{i+2} = u_i + u_{i+1}$ for all $i\}$ My question is: how can i ... linear-algebra sequences-and-series lucas-numbers- 288
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 \... sequences-and-series number-theory prime-numbers lucas-numbers- 12.7k
Does $(L_{p+1}+2)\equiv0 \mod p$ only when $p^2$ have digits in nondecreasing order?
Let $L_n$ be the $n$th Lucas number and $p$ a prime number. I noticed something with Lucas Number : it seems than $(L_{n+1}+2)\equiv0 \mod n$ is right only when $n$ is a prime $p$ and only if $p^2$ ... number-theory prime-numbers fibonacci-numbers lucas-numbers- 313
When is $L_n-1$ a prime?
Let $L_n$ be the $n$th Lucas number. I tested whether $L_n-1$ is prime for all $n<100000$ and found that it is prime only for $n=2,3,6,24,48,96$. Are there any other prime numbers? Also, is there a ... prime-numbers fibonacci-numbers lucas-numbers- 1,193
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