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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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How to prove? (Do not use mathematical induction)

I would appreciate if somebody could help me with the following problem: Q: Show that $$f_1=f_2=1, f_{n+2}=f_{n+1}+f_{n}(n\in \mathbb{N})~~~~ \Rightarrow ~~~~f_1f_2+f_2f_3+f_3f_4+\cdots+f_{2n-1}f_{2n}=f_{2n}^2$$
Young
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Relation between cubes of Fibonacci numbers and even Fibonacci numbers

While recently revisiting an old Project Euler problem, which states: Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,…
Emily
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What is the name of Fibonacci variation when $F(n) = a\cdot F(n-1) + b\cdot F(n-2) + c$, were $c$ is a constant, and $a >0, \ b >0, \ c>0$

I am trying to write $\log(n)$ algorithm for the above. I don't know if there is a specific name for the Fibonacci variation when: $$F(n) = a\cdot F(n-1) + b\cdot F(n-2) + c$$ where: $a >0, \ b >0, \ c>0$ Could someone help me with the name of this…
user3038404
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Prove that $F_n$ is a multiple of $F_k$ given that n is a multiple of k, using the identity $F_{a+b+1}=F_{a+1}F_{b+1}+F_aF_b$

I haven't been able to get very far in the problem at all because I can't figure out how to change it to the terms of the identity. All I have for the problem is $F_{kx}$ = $F_n$ and y$F_k$=$F_n$. But I don't understand what this has to do with the…
eclare
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Fast calculation of Fibonacci numbers

I was wondering how can i find $n$-th Fibonacci number using the recurrence equation $F(n)=F(n-1)+F(n-2)$ if starting $a$,$b$ is given some different random number say $a=5$,$b=8$ than $c=a+b$ than it follows $5,8,13,21,34$. After googling, I came…
Shadab Sayeed
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How does Fibonacci recurrence simplify

I'm trying to better understand the Fibonacci recurrence. I understand that the closed form solution is: $$F_n=\frac{1}{\sqrt{5}}\bigg[\Big(\frac{1+\sqrt5}{2}\Big)^n-\Big(\frac{1-\sqrt5}{2}\Big)^n\bigg].$$ I read that is can simplified…
Camilo
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How do I prove/disprove that $F_n^2 + F_{n\pm 1}^2 = F_{2n\pm 1}$, where $F_n$ is the $n$th Fibonacci number?

I was looking at the Fibonacci sequence: $$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,\ldots, F_n : F_n + F_{n - 1} = F_{n + 1} \land F_0 = 0$$ Here I noticed something; $\qquad F_n^2 + F_{n\pm1}^2 = F_{2n\pm1} : n\in \mathbb{N}$ For example; $$\ \…
Mr Pie
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Is $F_8 = 21$ the last Fibonacci number that satisfies $F_n = 6F_a + F_b$?

We know that $F_n = 4F_{n-3} + F_{n-6} = 11F_{n-5} + F_{n-10}$ and there are countless other relationships between Fibonacci numbers. However I was examining $F_n = 6F_a + F_b$ where $a,b > 0$. It appears that the only solutions are $\mathbf{8}$…
Ranking Software
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Is there a 3d fibonacci sequence?

Is there 3d equivalent to the fibonacci sequence? I see pictures of the golden spiral and was wondering if there was a similar kind of sequence that would fill up 3 dimensional space. I was looking at the Romanesco broccoli, which is a 3d shape, but…
jveazey
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Adding subscripts

This is a stupid question. But I'm trying to solve a Fibonacci problem and just realized that I don't know how to manipulate them. For example why does $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$
ematth7
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geometric proof for fibonacci numbers identity with sum of two squares

Induction Proof: Fibonacci Numbers Identity with Sum of Two Squares The link above gives the induction proof does a geometric proof using the squares with Fibonacci length exist for this?
zebra1729
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Prove that for each Fibonacci number $f_{4n}$ is a multiple of $3$.

The Fibonacci numbers are defined as follows: $f_0 = 0$, $f_1 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n \ge 2$. Prove that for each $n \ge 0$, $f_{4n}$ is a multiple of $3$. I've tried to prove to by induction. So, my basis is $f(0)$, which is…
user2847624
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Prove the given property of the Fibonacci numbers

I found in one of the books I read a lot of interesting properties of fibonacci numbers and among others this one in particular: For all $n \in \mathbb N$, $F_{n+1} F_{n-1} - F_n^2 = (-1)^n$. I tried to use induction but at some point on induction…
Kyle
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Fibonacci sequence is periodic $\!\bmod k$

In my study I understand that the Fibonacci sequence mod $k$ is periodic, with period less than $k^2$. Can any one generalize this with good proof?
Gandhi
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Proof for subsets of Fibonacci numbers

Let $a(n)$ be the number of subsets $A$ of $\{1,2,...,n\}$ with the property that $A$ is either the empty set or $\forall k \in A ( k \geq |A|)$. How can I prove that $a(n) = F(n+2)$ and show that this also implies $\displaystyle F(n+1) =…
user111556
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