# 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 $F_1=1$.

2,083 questions
Filter by
Sorted by
Tagged with
35 views

### Fibonacci propertie formula proof [closed]

I wanted to proof this Fibonacci formula $f^2_{n+1}= 4f_nf_{n−1}+ f^2_{n−2}$ without using combinatorial interpretation or induction on n, but I still stuck with nothing much. Can anyone, please, ...
147 views

### Is every Nth Fibonacci number where N is a power of 5 or 12 divisible by N?

I was playing around with Fibonacci numbers divisible by their indexes (i.e. the $12$th Fibonacci number, $144$, is divisible by $12$) and found that this works when the index is a power of $5$ or $12$...
1 vote
72 views

### Similarity of Lifting the Exponent Lemma in Pell numbers

Pell number is a term of the sequence $\{P_n\}$ determined by a recurrence relation $$P_{n+2}=2P_{n+1}+P_n, P_0=0, P_1=1.$$ Let $v_p(x)$ be the $p$-adic valuation of an integer $x$ (the number of ...
• 145
1 vote
121 views

### Finding the formula for the Fibonacci numbers using Generating Functions

I am trying to derive the formula for the Fibonacci sequence. Here is my work, I am making a mistake somewhere, but I can't seem to find where it is. My answer is almost correct but the the final ...
• 89
1 vote
25 views

### For which lowest value of $a$ and $b$ does their fibonacci series include a given number $n$? [closed]

Note: I am trying to get any useful insight into this type of questions, a general gist or mindset which would be helpful in dealing with question regarding fibonacci sequence. I am not currently ...
31 views

• 238
1 vote
74 views

32 views

### Analytic continuation of the sum of the reciprocals of the $n$-bonacci sequences.

In a previous question, I asked for an approximation of the sum of the reciprocals of the Tribonacci numbers. So I was wondering if there is a function for the sum of the reciprocals of the $n$-...
• 3,574
38 views

• 195
68 views

### Prove that $\gcd(F_n,F_{n+3})= 1$ or $2$. [duplicate]

Prove that $\gcd(F_n,F_{n+3})=1$ or $2$ for $n\geq 2$. The excercise has a slightly confusing hint. It says "Let $d|\gcd(F_n,F_{n+3})$" and also asks to show $d|2$ which I understand. I don'...
1 vote
64 views

### Fibonacci 19^n is multiple of 19 a mathematical property or a unique event? [duplicate]

Considering the 0 as the first index of Fibonacci numbers, I observed that Fibonacci(19^n) is multiple of 19. (Practically I could try till n=7). Is this a mathematical property of prime numbers or a ...
• 107
82 views

### How can I prove by induction that $3 ∣ f _{4n}$ is true?

This is related to the fibonaci numbers. I understand the way a proof by induction works, and these are 2 statements that are given: $f_1=1$ $f_2=1$ $f_n = f_{n-1} + f_{n-2}$, as long as $f\ge3$ (...
• 11