Questions tagged [fibonacci-numbers]

Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

The $n$th Fibonacci number $F_n$ is defined recursively, by

$$F_n = F_{n - 1} + F_{n - 2}$$

for $n > 1$, and $F_0 = 0,\; F_1 = 1$. There is a closed form expression, namely

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}}$$

where the golden ratio $\varphi$ is equal to $\frac{1 + \sqrt{5}}{2}$.

Combinatorial identities involving the Fibonacci numbers have been extensively studied, and the numbers arise frequently in nature and in popular culture.

Reference: Fibonacci number.

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Fibonacci Sequence (conceptual/mechanical)

I'm in Calc II and have my exam on series/sequences tomorrow. There will be an extra credit question related to Fibonacci and am trying to gather as much info on the sequence as possible. -If the Fibonacci sequence is increasing, and the ratio…
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Proving that a Fibonacci number is divisible by integer a

I am working on a review problem and can't figure out how to go about getting to an answer. We are told to let $F_n$ be the $nth$ Fibonacci number (defined as $F_1=F_2=1,F_{n+1}=F_n+F_{n-1}$). Show that, for any positive integer a, there is some…
user221400
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if d divides n then prove that fibonacci of d divides fibonacci of n

prove that if $d$ divides $n$ then prove that fibonacci of $d$ divides fibonacci of $n$. i have tried to write $F(n)$ as a multiple of $F(d)$ using the fact that $n = ad$ for some natural $a$ but got nowhere..
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What are the next few "tetranacci-like" pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the primes and the Lucas pseudoprimes, $$n = 705, 2465,…
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Prove that the sum of the first $n$ even terms of the Fibonacci sequence is given by $(F_{3n} + F_{3n + 3} - 2) / 4$ not using induction?

each even term in the Fibonacci sequence has a position index which is a multiple of 3, therefore the even term is $F_{3n}$: $F_{3 (0)} = 0$ $F_{3 (1)} = 2$ $F_{3 (2)} = 8$ $F_{3 (3)} = 34$ $F_{3 (4)} = 144$ ... How can I prove and reconstruct the…
user3019105
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Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015

This is a problem that has haunted me for more than a month. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind: Assume that the sequence $\{F_{n}\}$ of Fibonacci numbers is defined by the…
user237685
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"Non-commutative" Recurrence relation $ C_m = \alpha_m C_{m-1} + \beta_m C_{m-2} $

I have a problem, which is probably quite trivial. Consider a recurrence relation of the form $$ C_m = \alpha_m C_{m-1} + \beta_m C_{m-2}, $$ where the coefficients $\alpha_m$ and $\beta_m$ are non-commuting. In my problem, the $C_m$'s are functions…
Heidar
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Prove the following recurrence: $F_{2n+1}=3F_{2n-1}-F_{2n-3}$

Prove the following identity: $$F_{2n+1}=3F_{2n-1}-F_{2n-3}$$ So far I know that $F_n=F_{n-1}-F_{n-2}\implies F_{2n+1}=F_{2n}+F_{2n-1}$ Just not sure where to go from here to get to the conclusion. Note: $F_i$ is the $i^{\textrm{th}}$ term of the…
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Hadamard's product of Fibonacci generating functions.

$F(s) = \frac{1}{1-s-s^2}=\sum_{n\geq0}F_ns^n$. I want to calculate $F(s) \circ F(s) = \sum_{n\geq0}F_{n}^2s^n$. I have tried using Binet"s formula, but problem remains unsolved.
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Proof involving fibonacci number: $ \binom{n}{1}F_1+\binom{n}{2}F_2+\binom{n}{3}F_3+\cdots+\binom{n}{n-1}F_{n-1}+F_n=F_{2n}, $

Problem: Prove that $$ \binom{n}{1}F_1+\binom{n}{2}F_2+\binom{n}{3}F_3+\cdots+\binom{n}{n-1}F_{n-1}+F_n=F_{2n}, $$ where $F_n$ denotes the $n$th Fibonacci number. I tried induction, but I didn't know what to do for $n=k+1$.
abcde
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Convergence of fibonacci quotient $\frac{f_n}{f_{n+1}}$

I know that $\frac{f_{n+1}}{f_{n}}$ converges against $\phi = \frac{1+\sqrt{5}}{2}$. The question i want to to ask you is if the following conclusion is correct, I mean i know that if we have two convergent sequences in the (de-)nominator we can…
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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. Prove: (i) $b_n=b_{n-1} + b_{n-2}$ for $ n \ge…
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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 $n = 1,2,\cdots,k$, we see $L_{k+1} = L_k + L_{k-1}$ from an…
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Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and this slightly different formula has taken me…
JOX
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Counting the sequences of coin flips that end HH after $n$ flips (a more efficient method?)

I figured out that for any given $n$ the number of sequences of heads and tails that satisfy the condition that HH wasn't flipped consecutively until flips $n-1$ and $n$ is equal to the $(n-1)$th Fibonacci number. I found this by making a tree whose…
Acemanhattan
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