Questions regarding functions defined recursively, such as the Fibonacci sequence.

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A polynomial sequence

I have a sequence of polynomials $Q_k(x, y)$, $k\geq 1$ defined recursively as follows: $Q_1=x$. There is a sequence of polynomials $p_j(y)$ of degree $j$ such that $Q_{2m}$ is of the form \begin{...
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1answer
300 views

What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$?

Let $f_0=2$ and $f_n=f_{n-1}+\ln(f_{n-1})$. What is a good asymptotic to the sequence $f_n$? With good I mean much better than $f_n \sim \dfrac{3n \ln(2)\ln(n)}{2}$.
9
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1answer
389 views

How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
9
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1answer
332 views

Möbius function from random number sequence

Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it ...
9
votes
4answers
548 views

Help with a recurrence with even and odd terms [duplicate]

I have the following recurrence that I've been pounding on: $$ a(0)=1\\ a(1)=1\\ a(2)=2\\ a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n>1)\\ a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1) $$ I don't have much ...
9
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3answers
304 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove $$2^{m+2}|c_n\...
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1answer
82 views

A pair of sequences defined by mutual addition/multiplication

Define sequences $\{a_n\},\,\{b_n\}$ by mutual recurrence relations: $$a_0=b_0=1,\quad a_{n+1}=a_n+b_n,\quad b_{n+1}=a_n\cdot b_n.\tag1$$ The sequence $\{a_n\}$ begins: $$1,\,2,\,3,\,5,\,11,\,41,\,371,...
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1answer
339 views

Recurrence equation similar to a geometric progression

I have the following recurrence relation: $$T(i) = \sqrt{T(i-1) \left(T(i+1) + k\right)},$$ with $k \geq 0$, a fixed constant. I know that when $k=0$, we have: $$T(i) = \sqrt{T(i-1) T(i+1)},$$ which ...
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0answers
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How to Find the Recurrence Formula for $\int \frac{dx} {(1+\sin x)^n}$?

$$\int \frac{dx} {(1+\sin x)^n}$$ I know that I should use the following step: $$\int\frac{\sin^2(x)\, dx}{(1+\sin x)^n} +\int\frac{\cos^2(x)\, dx} {(1+\sin x)^n} $$ but here I get stuck. I do ...
9
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1answer
67 views

Prime Concatenation Order

Consider the following procedure. Given an integer $n \geq 2$, obtain the canonical prime factorization of $n$, i.e. $\prod_{i=1}^k p_i^{e_i}$. Take the distinct factors $p_i$ and list them in ...
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0answers
137 views

Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
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4answers
7k views

How to find the closed form formula for this recurrence relation

$ x_{0} = 5 $ $ x_{n} = 2x_{n-1} + 9(5^{n-1})$ I have computed: $x_{0} = 5, x_{1} = 19, x_{2} = 83, x_{3} = 391, x_{4} = 1907$, but cannot see any pattern for the general $n^{th}$ term.
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4answers
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How to get the characteristic equation?

In my book, this succession defined by recurrence is presented: $$U_n=3U_{n-1}-U_{n-3}$$ And it says that the characteristic equation of such is: $$x^3=3x^2-1$$ Honestly, I don't understand how. ...
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2answers
644 views

Solving recurrence relations that involve all previous terms

I'm not sure if this a proper recurance relation per se but I'd be interested in the methodology in solving a recurrence relation of the following form: $Z_0 = 1$ $Z_1 = x_1$ $Z_2 = x_1Z_1 + x_2 = ...
8
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2answers
675 views

Reduction formula for $I_{n}=\int {\cos{nx} \over \cos{x}}\rm{d}x$

What would be a simple method to compute a reduction formula for the following? $\displaystyle I_{n}=\int {\cos{nx} \over \cos{x}} \rm{d}x~$ where $n$ is a positive integer I understand that it ...
8
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1answer
273 views

How to solve $ \sqrt{x^2 +\sqrt{4x^2 +\sqrt{16x^2+ \sqrt{64x^2+\dotsb} } } } =5\,$?

How to find $x$ in: $$ \sqrt{x^2 +\sqrt{4x^2 +\sqrt{16x^2+ \sqrt{64x^2+\dotsb} } } } =5 $$
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1answer
1k views

Methods to find the limit of a sequence defined by a recurrence

For a sequence defined by a formula normally the usual limit rules allows one to find its limit. But for a sequence defined by a recurrence, up to now I have only seen some refined ad hoc methods, ...
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2answers
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Numerical method for finding the square-root.

I found a picture of Evan O'Dorney's winning project that gained him first place in the Intel Science talent search. He proposed a numerical method to find the square root, that gained him $100,000 ...
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2answers
246 views

Limit of the sequence $a_{n+1}=\frac{1}{2} (a_n+\sqrt{\frac{a_n^2+b_n^2}{2}})$ - can't recognize the pattern

Consider the sequence: $$a_0=x,~~~b_0=y$$ $$a_{n+1}=\frac{1}{2} \left(a_n+\sqrt{\frac{a_n^2+b_n^2}{2}} \right),~b_{n+1}=\frac{1}{2} \left(b_n+\sqrt{\frac{a_n^2+b_n^2}{2}}\right)$$ $$\lim_{n \to \...
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4answers
637 views

Closed form for a non-linear recurrence

Does equation $a_n=\sqrt{a_{n-1}+6}$ with $a_1=6$ have a closed form? I've found no linearization method. Any suggestion or hint will be highly appreciated.
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2answers
3k views

Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt 5}{...
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2answers
408 views

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
8
votes
1answer
165 views

Does $a_0=0, a_1=1, a_{n+2}=2a_{n+1}-3a_n$ ever return to 1 or -1 for $n>3$?

The sequence in question is the Lucas or Generalized Fibonacci sequence A088137. It's easy to write down its generating function $\frac{x}{1-2x+3x^2}$ and an explicit formula $a_n = \frac{\sqrt{2}}{2}...
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3answers
849 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
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1answer
170 views

Finding closed form for a generating function with different powers of $x$ in parameter

I'm working on a math/programming puzzle that involves an integer series defined as having a recurrence relating values $a(n)$ to $a(\lfloor\frac{n}{2}\rfloor)$ and $a(\lfloor\frac{n}{4}\rfloor)$. ...
8
votes
3answers
310 views

Quadratic Equation Recurrence?

So I was playing around with the following: Let $f_0(x) = x^2 - bx + c$. If $f_n(x)$ has roots $p$ and $q$ with $p > q$, then let $f_{n+1}(x) = x^2 - px + q$. The recurrence relation is rather ...
8
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1answer
142 views

Prove that a holonomic (p-recursive) difference equation returns only integral values

Consider the recurrence given by $(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$ $a_0 = 1, a_1 = 3$. Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...
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1answer
86 views

Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square ...
8
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1answer
392 views

Challenging recurrence relation problem

I am starting out with the following: $$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = \sum_{c=0}^n g(x)^{f(x)-c}\lambda_{n,c}(x) $$ Therefore: $$ \frac{d^{n+1}}{dx^{n+1}}[g(x)^{f(x)}] = \sum_{c=0}^{n+1}g(x)^{f(...
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1answer
274 views

Find the general term of the sequence defined by $x_0 = 3, x_1 = 4$ and $x_{n+1} = x_{n-1}^2 - nx_n$

Question: Find the general term of the sequence defined by $x_0 = 3, x_1 = 4$ and $x_{n+1} = x_{n-1}^2 - nx_n$. Attempt: If I'm not mistaken this does not match any linear homogeneous pattern, nor ...
8
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1answer
157 views

What are the solutions for $a(n)$ and $b(n)$ when $a(n+1)=a(n)b(n)$ and $b(n+1)=a(n)+b(n)$?

If you have the following recurrence relations : $a(n+1)= a(n) b(n)$ $b(n+1)= a(n) + b(n) $ How do you find the form of $a(n)$ and $b(n)$ ? I suspect there isn't a closed form , but a infinit sum ...
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2answers
99 views

Closed form of a recursive relation

A sequence $\langle a_n\rangle$ is defined recursively by $a_1=0$, $a_2=1$ and for $n\ge 3$, $$a_n=\frac 12 na_{n-1}+\frac 12n(n-1)a_{n-2}+(-1)^n\left(1-\frac n2\right).$$ Find a closed form ...
8
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1answer
438 views

How to solve this recurrence

Solve the recurrence \[ f_{j,k}^{(l)} = \begin{cases} \left[j>k\right] j^{k-1}(j-k), &\qquad j=l \\ \\ \left[j>k+1\right] \sum_t \binom k t f_{j-1,k-t}^{(l)}, &\qquad j>l \end{cases} \...
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0answers
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On the convergence of a more complex iterated radical

After exploring Ramanujan's famed $$3=\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}} $$ and $$4=\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}},$$ both of which can be expressed more generally by $$x+n+a=\sqrt{ax+(n+a)^2+x\...
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6answers
421 views

Finding the $n$-th derivative of $f(x) =e^x \sin x$, solving the recurrence relation

I was given a homework assignment to find a closed solution for the nth derivative of the function: $f(x) = e^x \sin x$ So far I have been able to obtain the derivative as: $f^{(n)}(x) = e^x S_n \...
7
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5answers
232 views

How can I find the value of $a^n+b^n$, given the value of $a+b$, $ab$, and $n$?

I have been given the value of $a+b$ , $ab$ and $n$. I've to calculate the value of $a^n+b^n$. How can I do it? I would like to find out a general solution. Because the value of $n$ , $a+b$ and $ab$ ...
7
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3answers
349 views

How can I rewrite recursive function as single formula?

There is following recursive function $$ \begin{equation} a_n= \begin{cases} -1, & \text{if}\ n = 0 \\ 1, & \text{if}\ n = 1\\ 10a_{n-1}-21a_{n-2}, & \text{if}\ ...
7
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2answers
473 views

Recurrent sequence limit

Let $a_n$ be a sequence defined: $a_1=3; a_{n+1}=a_n^2-2$ We must find the limit: $$\lim_{n\to\infty}\frac{a_n}{a_1a_2...a_{n-1}}$$ My attempt The sequence is increasing and does not have an upper ...
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650 views

How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$

I am self-studying Discrete Mathematics, and I have two exercises to solve. Find a formula for the following recurrence relation: (translated from Portuguese) a) $a_{1}=3,a_{2}=5, a_{n+1}=3a_{n}-2a_{...
7
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3answers
587 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
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3answers
270 views

How can I find an explicit expression for this recursively defined sequence?

We define the sequence $(u_n)_{n=1}^\infty$by:$$u_{n+1}=1+\frac{1}{u_n}$$ How can I find the limit of this sequence as it goes to infinity? By induction, I can prove that it is bounded above and ...
7
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1answer
116 views

Is there any explicit formula for $x_n$?

Let $a$ be a positive real number and $(x_n)$ be the sequence given by $x_1>0,$ $$x_{n+1}=\dfrac{1}{2}\Big(x_n+\dfrac{a}{x_n}\Big).$$ We can prove that $x_n\to\sqrt{a}$ as $n\to\infty.$ My ...
7
votes
2answers
251 views

A recurrence relation problem: $\frac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \frac 1 n$

I need to solve this recurrence problem to find $a_n$ $\dfrac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \dfrac 1 n$ It is what I tried so far: $$\log (\dfrac {a_{n-1}.a_{n+1}} {a_n^2}) = \log(1 + \dfrac 1 n)$...
7
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2answers
290 views

Does anyone recognise this recursion sastisfied by the Bell numbers?

I derived a recursion $$B_n=\frac{1}{n}B_0+\frac{1}{n}\sum_{k=1}^{n-1}\left[\binom{n}{k}-(-1)^{n-k}\binom{n}{k-1}\right]B_k\tag{$*$}$$ which I know should be satisfied by the moments of the unit ...
7
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1answer
311 views

How prove that $x_1 = x_{2000}$ implies $x_2 \ne x_{1999}$, where $x_{n+2}=\frac{x_{n}x_{n+1}+5x^4_{n}}{x_{n}-x_{n+1}}$?

Let $x_{1},x_{2},\cdots$ be real numbers, such that for $n \ge 1$: $$x_{n+2}=\dfrac{x_{n}x_{n+1}+5x^4_{n}}{x_{n}-x_{n+1}}$$ If $x_{1}=x_{2000}$, prove that $x_{2}\neq x_{1999}$. my idea $$x_{n+2}x_{...
7
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5answers
446 views

The sum $\sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j$

A recent answer of mine to a question on Math Overflow includes the sum $$S(n,k,x) = \sum_{j=0}^n \binom{n}{j} \left\{ j \atop k \right\} x^j,$$ where $\left\{ j \atop k \right\}$ is a Stirling number ...
7
votes
6answers
123 views

If $a_{n+1}=\frac {a_n^2+5} {a_{n-1}}$ then $a_{n+1}=Sa_n+Ta_{n-1}$ for some $S,T\in \Bbb Z$.

Question Let $$a_{n+1}:=\frac {a_n^2+5} {a_{n-1}},\, a_0=2,a_1=3$$ Prove that there exists integers $S,T$ such that $a_{n+1}=Sa_n+Ta_{n-1}$. Attempt I calculated the first few values of $...
7
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3answers
470 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
7
votes
4answers
391 views

How can I solve this linear recurrence relation?

My problem is: this given recurrence relation: $$y_{n+1}-\frac{n+2}{2}\cdot y_n = (n+1)(n+2)\cdot 3^n$$ for all: $n\ge 0$ and $y_0 = 0$ I need to find the explicit form and the general solution. My ...
7
votes
4answers
319 views

Limit of $x_n^3/n^2$ when $x_{n+1}=x_n+ 1/\sqrt {x_n}$ with $x_0 \gt 0$

Let $(x_n)_{n \ge 0}$ a sequence of real numbers with $x_0 \gt 0$ and $x_{n+1}=x_n+ \frac {1}{\sqrt {x_n}}$. Check the existence and find $$L=\lim_{n \rightarrow \infty} \frac {x_n^3} {n^2}$$ ...