Tagged Questions
1
vote
0answers
25 views
What is the broader name for fibonacci and lucas sequences?
Fibonacci and Lucas sequences are very similar in their definition. However, I could just as easily make another series with a similar definition; an example would be:
$$x_0 = 53$$
$$x_1 = 62$$
$$x_n ...
1
vote
3answers
56 views
Fibonacci series in 0-Even-Odd-Even-Odd-N series up to N
Another question from the test for the Normale of Pisa:
Consider the series $S_n$ of integer numbers repeteandly even - odd - even - odd that start with 0 and finish with n, so with n = 3 we get 2 ...
5
votes
4answers
173 views
The generating function for the Fibonacci numbers
$$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+...=\frac{1}{1-(z+z^2)}$$
The coefficients are Fibonacci numbers $\left\{1,1,3,5,8,13,21,...\right\}$.
Please HELP. Thanks guys.
1
vote
0answers
29 views
Strange equality of the operator E($Eu_n=u_{n+1}$)
The operator $E$ is defined as $Eu_n=u_{n+1}$.
I encountered a strange equality. when I tried out
Let $u_n$ represent a series such that
$$u_{n+2}=u_{n+1}+u_n. \tag{$\star$}$$
Or
...
14
votes
1answer
176 views
Fibonacci Sequence in $\mathbb Z_n$.
Consider a Fibonacci sequence, except in $\mathbb Z_n$ instead of $\mathbb Z$:
$$F(1) = F(2) = 1$$ $$F(n+2) = F(n+1) + F(n)$$
It is easy to see that each of these sequences must cycle through some ...
1
vote
1answer
38 views
quick approximation for largest fibonacci under a limit?
I asked in a previous post about finding a closed form for: $$\sum_{i=0}^{n}F_{3i}$$ which is the sum of the even fibs less than or equal to the nth even fib. the great answersI got showed me a very ...
5
votes
3answers
147 views
Closed form for the sum of even fibonacci numbers?
I recently took a look a project euler, and I am trying to think of a smart way to do number 2. I looked at the sequence, and I saw that the question is basically asking for
$$
\sum_{i=1}^n F_{3i}
$$
...
1
vote
1answer
45 views
All sequences constructed by using the denominator as nominator and the sum of denominator and nominator as denominator converges to $\phi-1$
Assume we are given any number a. Write it in the form $a = \frac{b}{c}$ (if rational, in the usual way, if irrational, use forms like $\frac{a}{1}$).
Construct a sequence ...
6
votes
1answer
116 views
Fibonacci numbers that are powers?
The Fibonacci sequence is:
$$\left(f_n\right) = \left(0,1,1,2,3,5,8,13,21,34,55,89,144,\dots\right)$$
where we start with $0$ and $1$ and each term in the sequence is the sum of the two previous ...
2
votes
2answers
101 views
Proof That the Ratio of Sucessive N-nacci Numbers Tends to 2.
For the paper that I'm working on involving N-nacci Recursion Formulas I need to prove that $$ \lim_ {n \to \infty}\lim_{a \to \infty} \left | \frac{f^{(n)}_{a+1}}{f^{(n)}_a} \right | = 2$$
To start ...
3
votes
1answer
78 views
Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{n}{f_n}$ where $f_n$ is the $n$'th Fibonacci number
Can we show convergence of$$B=\sum_{n=1}^{\infty}(-1)^n\frac{n}{f_n}$$where $f_n$ is the $n$'th Fibonacci number?
And then can we determine the exact value of $B$?
4
votes
1answer
79 views
Fibonnaci Sequence and series limits
1) Let $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3$.
How do you prove that
$$\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}} ...
2
votes
2answers
41 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, ...
5
votes
2answers
173 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:
$$A = \begin{bmatrix} 1&1 \\\\1&0\end{bmatrix}^n$$
...
1
vote
1answer
91 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 ...
1
vote
1answer
190 views
How to remove the denominator?
I have the following expression for $n>3$:
$$\frac{5\cdot(n-1)\cdot[8\cdot\operatorname{Luc}(n) + 5\cdot\operatorname{Luc}(n-1)] + [4\cdot\operatorname{Luc}(n-1) - ...
4
votes
1answer
231 views
Summation Of Product Of Fibonacci Numbers
Im trying to find out a general term for the following summation of products of fibonacci numbers:--
$$\sum_{k=4}^{n+1} F_{k}F_{n+5-k}\; , n \geq 3$$
I tried using Binet's equation but I am ...
0
votes
1answer
128 views
Reducing this expression to simpler form
$\newcommand{\Fib}{\operatorname{Fib}}$I am trying to reduce this expression for the $n$th term of sequence $G$.
$G[n]=\Fib(4) \times \Fib(n-1) + \Fib(5) \times \Fib(n-2) + \Fib(6) \times \Fib(n-3)+ ...
0
votes
2answers
647 views
Summation of series of product of Fibonacci numbers
What is the sum of following product of Fibonacci numbers
$$\sum_{k=1}^{n-1} Fib(k)*Fib(n+3-k)$$
can anyone suggest only approach to find general term?
4
votes
1answer
561 views
Sum of product of Fibonacci numbers
I want to calculate the sum of product of Fibonacci number for a given $n$. That is, for given $n$, say
$$F_1 F_n + F_2 F_{n-1} + F_3 F_{n-2} + F_4 F_{n-3} + F_5 F_{n-4} + \cdots.$$
what should be ...
4
votes
1answer
148 views
Two sums with Fibonacci numbers
Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$
Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ ...
15
votes
1answer
228 views
The Fibonacci sum $\sum_{n=0}^\infty \frac{1}{F_{2^n}}$ generalized
The evaluation,
$$\sum_{n=0}^\infty \frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}=\left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2$$
was recently asked in a post by Chris here.
I ...
8
votes
3answers
192 views
Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
Evaluate the sum:
$$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$
where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.
4
votes
1answer
234 views
What is the next “Tribonacci-like” pseudoprime?
Given the three roots of $x^3=x^2+x+1$. Then we get the tribonacci-like sequence,
$B_n = x_1^n+x_2^n+x_3^n = 3, 1, 3, 7, 11, 21, 39, 71, 131,\dots$
where $B_n = B_{n-1}+B_{n-2}+B_{n-3}$, and the ...
10
votes
1answer
242 views
Why do the Fibonacci numbers recycle these formulas?
The Fibonacci numbers $F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$
obey the following recurrence relations,
$ \begin{aligned}
&F_{n}-\;F_{n-1}-F_{n-2} = 0\\[1.5mm]
&F_{n-1}^3-F_{n}^3-F_{n+1}^3 = ...
2
votes
2answers
239 views
Use of the Reciprocal Fibonacci constant?
The Reciprocal Fibonacci constant ($\psi$) is defined as
$$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$
where $F_{k}$ is the $k^{th}$ Fibonacci number.
The irrationality of $\psi$ has been proven. ...
0
votes
2answers
119 views
A game: Fibonacci sequences and probability.
Let's play a game. You have two biased coins: coin A has a $0.4$ and $0.6$ for H and T probability and coin B has the opposite ($0.6$ and $0.4$ for H and T). These coins must be flipped one at a time, ...
6
votes
2answers
162 views
Prove that $\sum\limits_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= \sum\limits_{n=0}^{\infty}\frac{1}{2^{n}}$
I came up with this identity in high school, and I can't remember how I proved it :P Does anyone know how I would go about doing this?
$$\sum_{n=0}^{\infty}\frac{F_{n}}{2^{n}}= ...
3
votes
4answers
209 views
Showing that $f_{2n+1}=f_{n+1}^2+f_n^2$.
I am trying to solve the following exercise:
Let $f_1=1$, $f_2=1$, $f_{n+1}=f_n+f_{n-1}$, where $n\in\mathbb{N}$. Show that $f_{2n+1}=f_{n+1}^2+f_n^2$.
I have not had much progress, but this is ...
2
votes
3answers
191 views
Another way to go about proving Binet's Formula
As I showed in another question of mine, it is easy to prove that
$$\tag{1}\phi^{n+1} =F_{n+1} \phi+F_{n }$$
given $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}\text{ ; }n\geq2$.
Now, extending $(1)$ ...
8
votes
2answers
292 views
Infinite Series: Fibonacci/ $2^n$
I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner)
In the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... each term ...
7
votes
1answer
316 views
Another way to go about proving the limit of Fibonacci's sequence quotient.
It is not difficult to inductively prove that
$$\eqalign{
& \phi = \phi + 0 \cr
& {\phi ^2} = \phi + 1 \cr
& {\phi ^3} = 2\phi + 1 \cr
& {\phi ^4} = 3\phi + 2 ...
7
votes
3answers
151 views
How can I show that $\sum\limits_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$ is algebraic?
Show that $$\sum_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$$ is algebraic.
More specifically, solve this and get exact values.
Then use the result to evaluate $$\sum_{n=0}^\infty ...
11
votes
2answers
245 views
Generalized Fibonacci Sequence Question
The Fibonacci Sequence is defined as the recurrence
$a(n)=a_{n-1}+a_{n-2}$ where $a(0)=0$ and $a(1)=1$. Today, I was bored so I considered the sequence $a(n)=\sqrt{a_{n-1}}+\sqrt{a_{n-2}}$. Ten ...
7
votes
8answers
593 views
Need help deriving recurrence relation for even-valued Fibonacci numbers.
That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$
Empirically one can check that:
$a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$.
If $f(n)$ is ...
1
vote
4answers
220 views
Generalized Fibonacci sequences
Why Fibonacci sequence start at $0$, Tribonacci sequence with $0,0$, Tetranacci with $0,0,0$, etc. [ref OEIS] Has any good reasons for that?
These sequences arise in generalization of Pascal Triangle ...
2
votes
2answers
343 views
Recurrence for a lagged Fibonacci sequence
I know how to do the matrix for the standard $f(n) = f(n-1) + f(n-2)$ relationship, but what if it's piecewise?
For instance, $f(1)$ through $f(30)$ have some preset values, and then for $f(31)$ ...
11
votes
3answers
466 views
Prove the Fibonacci sum $\sum \limits_{n=0}^{\infty}\frac{F_n}{p^n} = \frac{p}{p^2-p-1}$
We are familiar with the nifty fact that given the Fibonacci series $F_n = 0, 1, 1, 2, 3, 5, 8,\dots$ then $0.0112358\dots\approx 1/89$. In fact,
$$\sum_{n=0}^{\infty}\frac{F_n}{10^n} = ...
3
votes
1answer
195 views
Expanding the generating function of the Fibonacci numbers to find a cute formula
$F_0=1$, $F_1=1$, $F_n=F_{n-1}+F_{n-2}$. The generating function is $-\frac{1}{x^2+x-1}$. I have to expand it to prove that $F_n=\sum_k\binom{k}{n-k}$. Could you help me please?
3
votes
1answer
366 views
On the generating function of the Fibonacci numbers
Let's define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be ...
23
votes
2answers
974 views
Why do some Fibonacci numbers appear in an approximation for $e^{\pi\sqrt{163}}$?
It is rather well-known that,
$e^{\pi\sqrt{43}} \approx 960^3 + 743.999\ldots$
$e^{\pi\sqrt{67}} \approx 5280^3 + 743.99999\ldots$
$e^{\pi\sqrt{163}} \approx 640320^3 + 743.999999999999\ldots$
Not ...
3
votes
3answers
96 views
Write $\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$ in a simpler form, where $F_n$ is the n-th element of the Fibonacci sequence?
The exercise asks to express the following:
$\sum_{1}^{n} F_{2n-1} \cdot F_{2n}$
in a simpler form, not necessarily a closed one. The previous problem in the set was the same, with a different ...
6
votes
3answers
812 views
Interesting properties of Fibonacci-like sequences?
Everyone is familiar with the Fibonacci Sequence, [0] 1 1 2 3 5 8 ... and many of it's interesting properties. For example, as the sequence continues, the ratio of ...
6
votes
1answer
209 views
Common terms in general Fibonacci sequences
Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
7
votes
1answer
218 views
Prove: the intersection of Fibonacci sequence and Mersenne sequence is just $\{1,3\}$
$$\frac{{{\varphi ^n} - {{(1 - \varphi )}^n}}}{{\sqrt 5 }} = {2^m} - 1 .$$
Here $\varphi = \frac{{1 + \sqrt 5 }}{2}$ . This integer equation has no solution for $n>3$ and $m>2$. How to prove?
