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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7
votes
2answers
57 views

A number $N$ is a $k$-nacci number if and only if …

For $k\ge 2\in\mathbb N$, one can define the $n$-th $k$-nacci number $f_k(n)\ (n=0,1,\cdots)$ as $$f_k(0)=f_k(1)=\cdots=f_{k}(k-2)=0,\ \ ...
2
votes
1answer
15 views

How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
-5
votes
1answer
16 views

is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$ and vice versa? [on hold]

Is there constant $k$ such that nth fibonnaci number $F_n$ satisfies $F_n > k2^n$? Also is there constant $k$ that $k2^n>F_n$?
-3
votes
1answer
27 views

…is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences. [on hold]

Let $$\begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*}$$ There is a unique ordered pair $(c,d)$ such that $c\phi^n + ...
1
vote
2answers
34 views

Limit and Fibonacci [on hold]

How to prove $${\lim_{n \to \infty} \frac{F_{kn}}{{F_n}^k} = 5^{(k-1)/2}}$$ Non-induction method is prefered.
1
vote
1answer
23 views

Are there 3D geometric proofs of Fibonacci identities?

There is a significant number of identities involving Fibonacci numbers that can be proven in a geometric way, like this one: However, I couldn't find any that involves 3D geometry. I also couldn't ...
0
votes
0answers
59 views

An identity for the Fibonacci number $F_{n^2}$

I was manipulating Fibonacci numbers defined by : $F_0=0$ and $F_1=1$ $ \forall n\in \mathbb{N}$ $F_{n+2}=F_{n+1}+F_n$ Until I obtain this equation (which I proved) $\forall n\in \mathbb{N^*}$: ...
4
votes
1answer
44 views

Closed form as sum and combinatorial of Fibonacci numbers

How can I prove that the Fibonacci numbers that are defined as $F_n=F_{n-1}+F_{n-2}, \; n \geq 2$ and $F_0=0,\ F_1=1,\ F_2=1$ have the form: $$F_n=\sum_{k=0}^{n-1} \binom{n-1-k}{k}, \; n\ge 2 $$ I ...
3
votes
0answers
131 views

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$?

What is a better comparision series for $\sum \frac {1}{F_n}$ than $\sum_0^\infty2^{-k}$? By better comparison series than $\sum_0^\infty2^{-k}$ we mean a series $\sum c_k$ s.t. ...
1
vote
1answer
42 views

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 = ...
2
votes
4answers
77 views

A calculus proof for the general term of the Fibonacci sequence

Let $a_0=1$,$a_1=1$ and $a_n=a_{n-1} + a_{n-2}$ for $n \geq 2$, I would like to prove: $$a_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n + 1}- \left(\frac{1-\sqrt{5}}{2}\right)^{n + ...
3
votes
4answers
42 views

Prove $F^2_{n+1} - F_nF_{n+2} = (-1)^n$

This is a question about Fibonacci sequences, a sequence in which the previous terms build up upon the current term (e.g. $F_1 + F_2 = F_3$ where $F_1 = F_2 = 1$). How would I go about proving ...
0
votes
1answer
33 views

are there infinitely many primes in Fibonacci sequence

There is one proof about infinitude of prime with following method, http://www.ams.org/mathscinet-getitem?mr=2271540 Also it is well know that any two consecutive Fibonacci numbers are mutually ...
2
votes
4answers
123 views

Prove that $p$ divides $F_{p-1}+F_{p+1}-1$ [duplicate]

Given the Fibonacci sequence $(F_n)$, defined by $F_0=0,F_1=1, F_{n+2}=F_{n+1}+F_n$, and $p$ an odd prime number, how to prove that $p$ divides $F_{p-1}+F_{p+1}-1$? Is induction a good idea here? ...
2
votes
2answers
67 views

Sum of squares of Fibonacci Numbers

$$ \sum_{i=0}^{n} (F_{2i+1})^2 = \;?$$ I know that sum of squares of first $n$ Fibonacci numbers is $F_{n} \times F_{n+1}$.
0
votes
2answers
40 views

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 ...
1
vote
1answer
35 views

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 ...
1
vote
4answers
51 views

Fibonacci sequences and related series

Let $\{a_n\}$ be a sequence such that $a_1=a_2=1$ and $a_{n+1}=a_n+a_{n-1}$ for $n\geq 2$. Prove that $\displaystyle \sum_{n=1}^\infty \frac{1}{a_n}$ converges. My work: Let $b_n=\frac{1}{a_n}$. ...
10
votes
4answers
206 views

Linear Combinations of Fibonacci Numbers (integer coefficients)

While working on problem #2 on Project Euler, I came across the need to express $F_n$ as a linear combination of $F_{n-3}$ and $F_{n-6}$. This is relatively simple to do: $$\begin{align} F_n &= ...
31
votes
6answers
5k views

Proof the Fibonacci numbers are not a polynomial.

I was asked a while ago to prove there is no polynomial $P$ in $\mathbb R$ such that $P(i)=f_i$ for all $i\geq0$. I tried to get a proof as slick as possible and here's what I got. Let ...
1
vote
2answers
36 views

Modulus of sum of sequence of Fibonacci numbers

What is the most efficient way to find the modulus of sum of sequence of fibinacci numbers. For example (F(N) + F(N + 1) + ... + F(M)) mod 1000000007.
15
votes
5answers
3k views

Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?

About this Fibonacci sequence, is it true that the remainders when divided by three repeat along with the sequence like this: Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
34
votes
1answer
1k views

Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
43
votes
3answers
505 views

Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.

Using a symbolic computation software (Mathematica), I got the following interesting results: $$ \begin{align} \sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{1+\frac{1}{1+n^2}}}} &= ...
0
votes
0answers
68 views

Hyper sum of Fibonacci numbers

Let $F(n)$ be the $n$-th Fibonacci number. That is, $F(n)$ satisfies $F(0)=0,F(1)=1,F(n)=F(n−1)+F(n−2) (if n≥2).$ Let $f_k(n)$ be the function such that $f_0(n) = F(n)$, $f_k(n) = ...
1
vote
2answers
83 views

Divisibility of $987x^n − F_nx^{16} + F_{n−16}$

If $F_n$ is $n^{th}$ Fibonacci number, and polynomials $P_n(x)$ are defined as $987x^n − F_nx^{16} + F_{n−16}$, prove that for all $n ≥ 1$, $P_n(x)$ is divisible by $x^2−x−1$. This is from a ...
19
votes
6answers
264 views

Why does every “fibonacci like” series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that ...
3
votes
0answers
36 views

Proving generalized Cassini's identity using determinant?

Motivation It is not hard to show, by using the general solution, that Proposition. If $(a_{n})_{n\in\Bbb{Z}}$ satisfies the recursive formula $ a_{n+2} = pa_{n+1} + qa_{n}$, then for any $n, i, ...
1
vote
0answers
21 views

Fibonacci, prove that $F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$ with induction [duplicate]

I need to prove by induction that: $$F_{n}\cdot F_{n+2}-({F_{n+1}})^2=(-1)^n$$ I did the following: Check if the statement holds for $n=1$: $$1\cdot 3-(2)^2=(-1)^1$$ Check if the statement ...
18
votes
1answer
378 views

Fibonacci-related sum

Related to this question Find a solution for f(1/x)+f(1+x)=x, what is this sum: $$\sum_{n=1}^{\infty}(-1)^n\left(\frac{F_n}{F_{n+1}}-\frac1{\phi}\right)$$ where $F_n$ is the $n$th Fibonacci number and ...
1
vote
1answer
38 views

Confusion on unberstanding the proof of induction regarding Fibonacci numbers

I am trying to understand the proof that "For all $n\geq 2, F_n^2-F_{n+1}F_{n-1}=(-1)^{n-1}$.Where $F_n$ stands for the Fibonacci number at $n$. I got this proof from a book and here is the proof. ...
0
votes
3answers
63 views

Matrices, determinants, and applications to identities involving Fibonacci numbers

Preamble It is well known that since: $$ \begin{pmatrix} F_{n+1} \\ F_n \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix} \begin{pmatrix} F_n & F_{n-1} ...
2
votes
2answers
59 views

Seven expressions involving $F_n$ an $L_n$ that are always composite

Prove that if $F_n$ an $L_n$ are Fibonacci and Lucas numbers respectively, and $n>2$, then $$F_{n-2}\times F_{n-1}\times F_{n+1}\times F_{n+2}-15$$ $$F_{n-2}\times F_{n-1}\times ...
6
votes
1answer
105 views

Prove or disprove that ${F_{n}}^2 + 41$ is always a composite

The problem: Prove or disprove: If $F_{n}$ is the $n^{th}$ Fibonacci number then $${F_{n}}^2 + 41$$ is always a composite number. It looks that if $n$ is not multiple of 12, ${F_{n}}^2 + 41$ ...
6
votes
1answer
95 views

Prove or disprove that ${F_{n}^2} + 43$ is always a composite

This is a kind of follow-up to another question, but in order not to burden that question and its answers with new comments, I decided to create this separate question. Also, it looks this problem is ...
32
votes
5answers
644 views

Is ${F_{n}}^2 - 28$ always a composite number?

The problem: Prove or disprove that if ${F_{n}}$ is $n$-th Fibonacci number, and $n>5$ $${F_{n}}^2 - 28$$ cannot be a prime. I came to this accidentally while trying to solve another ...
2
votes
0answers
80 views

Who First Considered This Generalization of the Fibonacci Numbers?

I am looking for the author who originally researched a generalization of the Fibonacci numbers, which Koshy, in Chapter 7 his book Fibonacci and Lucas Numbers with Application refers to as the ...
5
votes
1answer
112 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as: Lets say we have the number $p$ which is an odd prime. Prove that: $F_{p-1} + F_{p+1} -1$ Is divisible by $p$. Prove that for any given $n$ real ...
3
votes
1answer
70 views

Summation of a multiple series involving Fibonacci numbers

Compute the sum $$\sum_{a_{2015} = 0}^{\infty} \sum_{a_{2014} = 0}^{a_{2015}} \sum_{a_{2013} = 0}^{a_{2014}} \cdots \sum_{a_{1} = 0}^{a_2} \sum_{k=0}^{a_1} \frac{F_{k}}{2^{a_{2015}}} $$ where $F_k$ ...
6
votes
3answers
66 views

If $f_{n-1}^2=(f_n/2)^2+h^2$ then $n=6$

How can I prove that if $f_n$ is a term of the Fibonacci sequence divisible by $4$ and if $$f_{n-1}^2=(f_n/2)^2+h^2,$$ $h\in\Bbb Z^+$ then $n=6$? I know that since $\gcd(f_k,f_{k+1})=1$ for every ...
0
votes
0answers
40 views

Induction: Fibonacci / Lucas Numbers [duplicate]

From Andrews' Number Theory, Chapter 1, Section 1, Problem 15: Prove, by induction, that $F_{2n} = F_nL_n$ where $F_n$ denotes the nth Fibonacci number and $L_n$ denotes the nth Lucas ...
0
votes
2answers
83 views

Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number [duplicate]

If $F_n$ is the $n$-th fibonacci number, then prove that, $$F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$$ I tried the idea of using Pascal's triangle, but it seems to need some ...
1
vote
2answers
72 views

A series for Fibonacci numbers.

How can I prove The Fibonacci sequence is encoded in the number $1/89$ i.e. $( 1/89 = 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034 \ldots)$
4
votes
3answers
200 views

Inequality of the Fibonacci sequence and the golden ratio

How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. Any help is really appreciated
1
vote
2answers
43 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
2
votes
3answers
71 views

Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation

How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series representation (something neat similar to $\frac{1}{1-x}$)? This reminds me of the power series ...
1
vote
4answers
58 views

How to prove this equation by induction?

I am trying to prove this equation by mathematical induction $$f_{n+1}f_{n-1} = f_{n}^{2}+(-1)^n$$ is true where $f_{n} = $ the nth number in the Fibonacci sequence. I don't quite get how to do this ...
0
votes
2answers
24 views

Problem on deriving binet formula

I'm trying to understand binet formula. I got a good explanation here. Please look at the link. Everything just fine but one thing. It said that $A_n = A_{n-1} + A_{n-2}$, which is fibonacci. But why ...
2
votes
3answers
76 views

Showing convergence of recursive sequence $A_{n+1}=\frac 1 {1+A_n}$

Given : $\forall n\in\Bbb N,\quad A_{n+1} = \frac 1 {1+A_n}$ and $A_1 = 0$ Show the sequence converges and find its limit. Briefly what I did was to create two sub-sequences with an index ...
1
vote
1answer
39 views

Prove that the set of solutions to $F_{n+2} = F_{n+1} + F_n$ is of dimension 2

I was playing with the Fibonacci sequence, willing to prove that $$ F(n) = \frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right) $$ I did the usual, ...