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 $F_1=1$.

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3
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2answers
230 views

Another Fibonacci identity

Here's a problem that is leading me in circles. Consider the Fibonacci number $F_n$ defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 2$. Prove that $F_{2n-1} = F_{n}^2 ...
1
vote
0answers
155 views

The zeta-function of Fibonacci sequence?

I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some ...
5
votes
5answers
286 views

Fibonacci Proof

Prove that: $$F_1F_2+F_2F_3+F_3F_4+\cdots+F_{2n-1}F_{2n}=F_{2n}^2$$ I set it up so: $$F^2(2k) + F(2k+1)F(2k+2) = F^2(2k+2)$$ I've tried: $$F(2k)^2 + F(2k+1)*F(2k+2) = ...
7
votes
1answer
328 views

Irrationality of reciprocal Fibonacci constant

I read that it was proved that reciprocal Fibonacci constant $$\sum_{n} \frac{1}{F_n} = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \frac{1}{13} + \frac{1}{21} ...
2
votes
2answers
158 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 ...
8
votes
1answer
556 views

N-nacci Identities: The Final Question (Generalizing Time!)

Okay so here is my personal work on the problem set. I only have question 5 remaining which involves generalization of any recursive sequence. $n$'s correspond to the $n$ in n-nacci. I hope to write ...
0
votes
2answers
103 views

Finding $a \bmod b$ where $a,b$ are large Fibonacci numbers

For moderately large values of $b$ we can use Chinese Remainder Theorem, by factorizing $b$. But for very large values of $b$, (for example $b$ is the 1000th Fibonacci number) factorization will take ...
4
votes
4answers
966 views

Fibonacci( Binet's Formula Derivation)-Revised with work shown

Okay so here is the revised question with my current work. Links to previous post(s)(Just for Gerry): Fibonacci Numbers - Complex Analysis Here's my attempt on the problem set thus far: (Note ...
-2
votes
0answers
190 views

Fibonacci Numbers - Complex Analysis [duplicate]

Possible Duplicate: Complex Analysis - Integral over a circle of radius R Hey guys~ Does anyone know where to find the solutions to this problem set on page 106 involving the fibonacci ...
4
votes
1answer
101 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 $B$?
4
votes
2answers
167 views

$n +1$th Fibonacci number modulo $n$

The Pisano period studies the $n$th Fibonacci number $F_{n}$ modulo $n$. Is there anything about $F_{n + 1} \pmod n$?
4
votes
1answer
158 views

Fibonacci Sequence and series limits

1) Let $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$. How do you prove that $$\sum_{n=2}^\infty \frac1{F_{n-1} F_{n+1}} = 1$$ $$\sum_{n=2}^\infty \frac{F_n}{F_{n-1} F_{n+1}} ...
1
vote
1answer
300 views

Fibonacci Generating Function of a Complex Variable

So I'm doing work on the Fibonacci Numbers, and I came across this problem for the generating function for the recursive fibonacci numbers. I have two questions: 1. Why is it useful to use a ...
2
votes
1answer
787 views

Causal Inverse Z-Transform of Fibonacci

Say the Fibonacci sequence is defined by: $y(n) = y(n-1) + y(n-2)$ initial conditions: $y(0)=0, y(1)=1$ I incorporate those initial conditions as: $y(n) = y(n-1) + y(n-2) + \delta(n-1)$ ...
0
votes
2answers
109 views

number theory fibonacci

Using facts of the Fibonacci sequence, I need to show that if $m,n$ are natural numbers that satisfy $m \mid F_n$ and $m \mid F_{n+1}$, then $m=1$. I am not sure where to start with this.. I am ...
1
vote
4answers
145 views

Proving that $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$, where $\alpha$ is the golden ratio

I got stuck on this exercise. It is Theorem 1.15 on page 14 of Robbins' Beginning Number Theory, 2nd edition. Theorem 1.15. $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$. Proof: ...
1
vote
3answers
2k views

nth term in the fibonacci series

I have read in a page that " To find the nth term of the Fibonacci series, we can use Binet's Formula" F(n) = round( (Phi ^ n) / √5 ) provided n ≥ 0 where Phi=1·61803 39887 49894 84820 ...
3
votes
2answers
459 views

Fibonacci / Lucas Numbers Relationship: $F_{2n} = F_n L_n$

Prove the identity by induction: $$ F_{2n} = F_n L_n, $$ where $F_n$ and $L_n$ are the $n^{th}$ Fibonacci and Lucas number, respectively. I have an answer but am not happy with it since it doesn't ...
1
vote
2answers
192 views

Prove $F_{n+1}F_{n-1}-(F_{n})^2=(-1)^n$ without induction

I am asked to pove the statement about fibonacci sequence. The task is from the passage about series and sequences. But the proof seems to need induction way, doesn't it? Prove the statement ...
7
votes
3answers
591 views

Fibonacci numbers divisible by $9$

The $n$th Fibonacci number $F_n$ is defined as follows,$$F_1=F_2=1\mbox{ and } F_{n+2}=F_{n+1}+F_{n}\mbox{ for } n\geq 1$$ I want to know how many of the first $1000$ Fibonacci numbers are divisible ...
2
votes
2answers
80 views

Non-isomorphic combinatorial classes with growth rate equal to the golden ratio?

In a couple of weeks, I'm giving a talk about the growth rates of some combinatorial classes. In the introduction, I thought I'd present the class of tilings of a $n\!\times\!2$ strip with dominos, ...
4
votes
3answers
332 views

Why doesn't this induction “proof ”show $f_n = (\phi)^n + (1-\phi)^n$?

Here, $\phi$ is the golden ratio and $f_n$ is the $n^{th}$ Fibonacci number. The formula I'm using is actually the closed form of the Lucas numbers. Let $n = 1$. Then $f_n = 1$ and $\phi + 1 - \phi ...
3
votes
2answers
756 views

Induction Proof: Formula for Fibonacci Numbers as Odd and Even Piecewise Function

How can we prove by induction the following? $ F_{n+1} = \left\{ \begin{array}{l l} F_{n/2}^2+F_{(n+2)/2}^2 & \quad \text{if $n$ is even}\\ ...
4
votes
1answer
2k views

Induction Proof: Formula for Sum of n Fibonacci Numbers

I am stuck though on the way to prove this statement of fibonacci numbers by induction : my steps: definition: $F_{0}:=0, F_{1}:=1 $ and $F_{n}:=F_{n-1}+F_{n-2}$ The Hypothesis is: $\sum_{i=0}^{n} ...
3
votes
3answers
307 views

Is there a closed form equation for fibonacci(n) modulo m?

Basically I am curious if there's a direct way to calculate fibonacci(n) modulo m with a closed form formula so I don't have to bother with matrix exponentials.
0
votes
1answer
136 views

fibonacci question [duplicate]

Possible Duplicate: Recurrence relation, Fibonacci numbers $(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$. $(i)$ Assume that all $a_n$ are ...
1
vote
1answer
102 views

What does $F^2_n$ mean?

In this Wikipedia entry on Cassini's identity, I saw this equation: $F_{n-1}F_{n+1}-F^2_n=(-1)^n$ $F^2_n$, what does that mean? Is it a summation signs for n to 2? I don't know what it means.
2
votes
1answer
930 views

Recurrence relation, Fibonacci numbers

$(a)$ Consider the recurrence relation $a_{n+2}a_n = a^2 _{n+1} + 2$ with $a_1 = a_2 = 1$. $(i)$ Assume that all $a_n$ are integers. Prove that they are all odd and the integers $a_n$ and $a_{n+1}$ ...
2
votes
2answers
425 views

Fibonacci conjecture: $(F_{n+5})^2 - (F_n)^2 = 3((F_{n+3})^2 - (F_{n+2})^2) + 8 F_{n+2} F_{n+3} $.

So this is the question I have The Fibonacci sequence is a recurrence system given by $$F_1 = 1, \ F_2 = 1, \ F_{n+2} = F_{n+1} + F_n \qquad (n = 1, 2, 3, \ldots).$$ This question concerns the ...
2
votes
1answer
171 views

Another Bijective proof for Fibonacci Identities

I'm going through a past exam, and this question popped up: Prove: $3P_n = P_{n-2} + P_{n+2},\,n>2$ Where $\tilde{P}_j = \{\textrm{monomer-dimer pavings on a } 1\times j \textrm{ board}\}$ and ...
2
votes
1answer
246 views

Bijective Proof of a Fibonacci Identity

Prove (Using bijections): $F_{1}+F_{3}+\cdots+F_{2n-1}=F_{2n}$ Where $F_{i}$ is the $i$th Fibonacci number. Apparently you use monomers and dimers to prove this, but I don't really know what to ...
4
votes
2answers
119 views

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$?

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ? (i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)? Let the number ...
3
votes
1answer
171 views

Diophantine equation: fermat numbers and fibonacci numbers

My question is how to find all solutions $(m,n)\in\mathbb N^2$ for $F_n=f_m$, where $F_n=2^{2^n}+1$ and $f_m$ is the $m$th fibonacci number: $f_0=0$, $f_1=1$ and $f_n+f_{n+1}=f_{n+2}$ for each ...
3
votes
4answers
219 views

Explaining a Fibonacci

Explain why the number below is not 299th Fibonacci number: 222232244629420445529739893461909967206666939096499764990979600 I need an explanation
1
vote
2answers
385 views

Primitive squareful Fibonacci numbers

In https://oeis.org/A065069 numbers $n$ such that Fibonacci($n$) is not squarefree, but for all proper divisors $k$ of $n$, Fibonacci($k$) is squarefree, are listed. OEIS gives a Mathematica program ...
5
votes
4answers
3k views

How to prove gcd of consecutive Fibonacci numbers is 1? [duplicate]

Possible Duplicate: Prove that two any consecutive terms of Fibonacci sequence are relatively prime How to prove it ? Can you help me ? Let $f_n$ be Fibonacci Sequence. ...
2
votes
6answers
274 views

Fibonacci equality, proving it someway

$ F_{2n} = F_n(F_n+2F_{n-1}) $ $ F_n $ is a nth Fibonacci number. I tried by induction but i didn't get anywhere
6
votes
2answers
616 views

Computing nth term of fibonacci-like sequence for large n

Sum up to nth term of fibonacci sequence for very large n can be calculated in O($\log n$) time using the following approach: $$A = \begin{bmatrix} 1&1 \\\\1&0\end{bmatrix}^n$$ ...
0
votes
2answers
135 views

Fibonacci Matrix

I'm looking for a good way to find the matrix form of fibonacci equation, and also a more general implementation. I've looked all other the web and haven't found it. I'll be really thankfull to any ...
2
votes
1answer
231 views

Proof the following proposition: for all $n \geq 0, \mathrm{fib}(n) \leq n!$

I am a comp science undergrad and just started to learn proof. And I have been thinking about this question for a few days. How should I present my answer? Do I have to use the Binet's formula? Or can ...
2
votes
1answer
590 views

Proof about lucas numbers.

Define the Lucas numbers to be $$l_n = l_{n-1} + l_{n-2} $$ if $n \ge 2$ with initial conditions $l_0 = 2$ and $l_1= 1$. I "proved" by induction that $l_n = f_{n-1} + f_{n+1}$ for $n \ge 1$ (by ...
1
vote
2answers
99 views

Existence of 5-d centrally symmetric self-dual polytope

Does there exist a 5-dimensional centrally symmetric self-dual polytope?
1
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2answers
120 views

Proof for one of the Lucas problem

Can anybody provide a combinatorial proof or algebraic proof of following identity? $${n\choose 0 }+ {n-1\choose 1}+{n-2\choose 2}+ .. +{{n-\lfloor n/2\rfloor} \choose {\lfloor n/2\rfloor}} = ...
1
vote
1answer
137 views

How to solve for the $n$-th Fibonacci number that is greater than or equal to $N$?

The general formula for the $n$-th Fibonacci number is: $$\frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}$$ where $$\phi = \frac{1 + \sqrt{5}}{2}$$ Given $N$, is there a way to solve for $n$ in this ...
7
votes
2answers
3k views

Every natural number can be written as the sum of distinct Fibonacci numbers?

Can anyone hint me to prove: $\forall n\in \mathbb{N}: \exists$ Fibonacci numbers $ F_{i_1},\ldots,F_{i_k}$ such that: $$\sum F_{i_k}=n$$ Note: Every Fibonacci number can appear only once. Thanks
1
vote
1answer
124 views

Power Variant of Fibonacci sequence

I was trying to simplify the following sequence, such that I can calculate the $n$th term in $\log n$ time. This can be done, if we can express the $n$th in terms of combinations of Fibonacci like ...
1
vote
1answer
254 views

Need formula for sequence related to Lucas/Fibonacci numbers

I am trying to get a formula for the nth term of the following sequence: 2, 14, 22, 58, 266, 398, 1042, 4774, 7142, 18698, 85666, 128158, 335522,... It's not in OEIS and as far as I can tell ...
0
votes
1answer
1k views

How to solve tribonacci series [duplicate]

Possible Duplicate: Fibonacci, tribonacci and other similar sequences Suppose my Tribonacci series is like this: \begin{equation} T(n) = T(n-1) + T(n-2) +T(n-3) \end{equation} with initial ...
0
votes
2answers
244 views

How to reduce the complexity of calculating the following summation? [duplicate]

Possible Duplicate: Summation of series of product of Fibonacci numbers Find $$\sum_{i=1}^{N-1}F_{i + 1} F_{N + 4 - i}$$ Is there a direct formula to calculate this, rather than actually ...
1
vote
1answer
378 views

How to remove the denominator?

I have the following expression for $n>3$: $$\frac{5\cdot(n-1)\cdot[8\cdot\operatorname{Luc}(n) + 5\cdot\operatorname{Luc}(n-1)] + [4\cdot\operatorname{Luc}(n-1) - ...