# Tagged Questions

25 views

### Fibonacci series mod a number

I'm trying to write a program with an input of numbers $n$ and $k$ (where $n<10^{20}$ and $k<10^9$), where I compute fib[n] % k. What is a good FAST way of computing this? I have read many ...
46 views

### Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
140 views

### What is the sum of Fibonacci reciprocals?

How can I calculate $\sum\limits_{n=1}^{\infty}\frac{1}{F_n}$, where $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$? Empirically, the result is around $3.35988566$. Is there a "more mathematical way" to ...
61 views

106 views

### Is Fibonacci sequence the minimum of unique pairwise sum sequence?

Let $(a_n)_{n=1}^\infty$ be a strictly increasing (condition added per earlier answer of Amitesh Datta) sequence of natural numbers where all pairwise element sums are unique. Can anyone prove or ...
58 views

### Solving $F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i})$?

I need to find $F_{n}$ in : $$F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2$$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
74 views

88 views

### Fibonacci series in 0-Even-Odd-Even-Odd-N series up to N

Another question from the test for the Normale of Pisa: Consider the series $S_n$ of integer numbers repeteandly even - odd - even - odd that start with 0 and finish with n, so with n = 3 we get 2 ...
2k views

### The generating function for the Fibonacci numbers

$$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$ The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$. Please HELP. Thanks guys.
42 views

### Strange equality of the operator E($Eu_n=u_{n+1}$)

The operator $E$ is defined as $Eu_n=u_{n+1}$. I encountered a strange equality. when I tried out Let $u_n$ represent a series such that $$u_{n+2}=u_{n+1}+u_n. \tag{\star}$$ Or ...
231 views

### Fibonacci Sequence in $\mathbb Z_n$.

Consider a Fibonacci sequence, except in $\mathbb Z_n$ instead of $\mathbb Z$: $$F(1) = F(2) = 1$$ $$F(n+2) = F(n+1) + F(n)$$ It is easy to see that each of these sequences must cycle through some ...
73 views

### quick approximation for largest fibonacci under a limit?

I asked in a previous post about finding a closed form for: $$\sum_{i=0}^{n}F_{3i}$$ which is the sum of the even fibs less than or equal to the nth even fib. the great answersI got showed me a very ...
54 views

### All sequences constructed by using the denominator as nominator and the sum of denominator and nominator as denominator converges to $\phi-1$

Assume we are given any number a. Write it in the form $a = \frac{b}{c}$ (if rational, in the usual way, if irrational, use forms like $\frac{a}{1}$). Construct a sequence ...
220 views

### Fibonacci numbers that are powers?

The Fibonacci sequence is: $$\left(f_n\right) = \left(0,1,1,2,3,5,8,13,21,34,55,89,144,\dots\right)$$ where we start with $0$ and $1$ and each term in the sequence is the sum of the two previous ...
142 views

### Proof That the Ratio of Sucessive N-nacci Numbers Tends to 2.

For the paper that I'm working on involving N-nacci Recursion Formulas I need to prove that $$\lim_ {n \to \infty}\lim_{a \to \infty} \left | \frac{f^{(n)}_{a+1}}{f^{(n)}_a} \right | = 2$$ To start ...
91 views

### Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{n}{f_n}$ where $f_n$ is the $n$'th Fibonacci number

Can we show convergence of$$B=\sum_{n=1}^{\infty}(-1)^n\frac{n}{f_n}$$where $f_n$ is the $n$'th Fibonacci number? And then can we determine the exact value of $B$?
110 views

299 views

### Fibonacci nth term

It is known that the nth term of the Fibonacci sequence can be found by the formula: $F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$, where $\phi$ is the golden ratio (1.618...). Would this be the ...
Im trying to find out a general term for the following summation of products of fibonacci numbers:-- $$\sum_{k=4}^{n+1} F_{k}F_{n+5-k}\; , n \geq 3$$ I tried using Binet's equation but I am ...