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

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9
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1answer
292 views

Prove that this particular sequence contains an infinite number of sixes

Given the sequence $$2,7,1,4,7,4,2,8,\ldots$$ which begins with $2, 7$ and is constructed by multiplying successive pairs of its members and adjoining the results as the next one or two members of ...
9
votes
3answers
339 views

Obtaining binomial coefficients without “counting subsets” argument

I want to obtain the formula for binomial coefficients in the following way: Elementary ring theory shows that $(X+1)^n\in\mathbb Z[X]$ is a degree $n$ polynomial, for all $n\geq0$, so we can write ...
9
votes
1answer
380 views

Recurrence for the partition numbers

I'm reading Analytic Combinatorics [PDF] book by Flajolet and Sedgewick, and I can't figure out one of the steps in the derivation of the $P_n$ — number of partitions of size $n$ (or coefficients in ...
9
votes
2answers
97 views

Proving that $a_n$ is an integer for every $n$

For every $k\ge1$ integer number if we define the sequence : $a_1,a_2,a_3,...,$ in the form of :$$a_1=2$$ $$a_{n+1}=ka_n+\sqrt{(k^2-1)(a^2_n-4)}$$ For every $n=1,2,3,....$ how to prove that $a_n$ is ...
9
votes
1answer
341 views

Closed form for $a_{n+1} = (a_n)^2+\frac{1}{4}$

I've been given the following sequence: \begin{align*} &a_0 = 0; \\ &a_{n+1} = (a_n)^2+\frac{1}{4}. \end{align*} I also have to prove that whatever I come up with is correct, but that will ...
9
votes
1answer
254 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
votes
3answers
276 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 ...
9
votes
1answer
296 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 ...
9
votes
2answers
430 views

Exact probability of random graph being connected

The problem: I'm trying to find the probability of a random undirected graph being connected. I'm using the model $G(n,p)$, where there are at most $n(n-1) \over 2$ edges (no self-loops or duplicate ...
9
votes
0answers
254 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
8
votes
2answers
571 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
votes
2answers
546 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
votes
3answers
2k 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 ...
8
votes
2answers
1k views

Linear homogeneous recurrence relations with repeated roots; motivation behind looking for solutions of the form $nx^n$?

If we have a linear homogeneous recurrence relation, such as $t_{k+1}=4t_k-4t_{k-1}$, and attempt to find solutions of the form $t_n=x^n$ for some $x \in \mathbb{R} \setminus \{0\}$, we obtain the ...
8
votes
2answers
1k views

Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct?

Problem The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation: $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Evaluate the limit $\lim_{n\to\infty} a_n$. Solution Show that the sequence ...
8
votes
4answers
475 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.
8
votes
1answer
213 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 ...
8
votes
1answer
127 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 ...
8
votes
1answer
417 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} ...
7
votes
8answers
819 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 ...
7
votes
4answers
3k 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.
7
votes
3answers
3k views

Why is solving non-linear recurrence relations “hopeless”?

I came across a non-linear recurrence relation I want to solve, and most of the places I look for help will say things like "it's hopeless to solve non-linear recurrence relations in general." Is ...
7
votes
3answers
1k views

Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?

1) Solve the recurrence relation $$T(n)=\begin{cases} 2T(n-1)+1,&\text{if }n>1\\ 1,&\text{if }n=1\;. \end{cases}$$ 2) Name a problem that also has such a recurrence relation. The ...
7
votes
6answers
382 views

Finding the $n$-th deriviative 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 deriviative 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
votes
5answers
452 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, ...
7
votes
1answer
306 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 ...
7
votes
4answers
278 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
2answers
594 views

Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...
7
votes
2answers
1k views

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 ...
7
votes
3answers
116 views

Help me understanding logic behind limits of recurence relations

I was trying to understand how limits of recurence relations are working. I have one. $$a_0 = \dfrac32 ,\ a_{n+1} = \frac{3}{4-a_n} $$ So, from what i know, if this recurence relation has a limit, ...
7
votes
1answer
217 views

Finding $a_n$ for very large $n$ where $a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $

I have a recurrence relation, $$ a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $$ for $n>3$ and $a_1 = 0, a_2 = 0, a_3 = 1$ I have to find the value of $a_n$ for very large values of n. I tried ...
7
votes
1answer
126 views

Put a mouse to the last cell

We have (n=12) cells $C_1, C_2 ,\dots, C_{12}$ which are initially empty. At each step, we can do one of two operations: $\mathbf{P}$: Put only in the first cell $C_1$ 2 mice. $\mathbf{M}$: Move ...
7
votes
2answers
255 views

q-Analogue of the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$.

The stirling numbers of the second kind satisfy the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$, where $(x)_k$ is the falling factorial. Consider the $q$-analog recursive definition of the ...
7
votes
1answer
302 views

How to prove this sequence of inequalities

The number $c_{g}(n)$ is defined by the recurrence \begin{equation} c_{g}(n) = c_{g}(n-1)+ (n-1)(n-2)c_{g-1}(n-2) , \end{equation} with $c_{0}(n)=1$ for any $n\geq 1$ and $c_{g}(n)=0$ if $n \leq 2g$. ...
7
votes
1answer
180 views

Diagonal of the double sequence $(n+1)v_{h,n+1}-(2h+1)v_{h,n}-nv_{h,n-1}=0$

Update: it is not possible to reply to this question without additional information. My comment below: "I have to agree with you that one "cannot derive (2) from (1) alone". Now it seems to me that ...
7
votes
1answer
358 views

Generating function of words in a binary alphabet counting blocks and appearances

Given the binary alphabet {a,b}, I'm trying to find the generating function that distinguishes, for all words of fixed length $n$, the count of blocks of a's and the number of a's. Let $x^p$ count the ...
7
votes
1answer
44 views

Recurrence $u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$

I was going through some old puzzles and encountered one where there was a non-linear recurrence of form $$u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$$ with $u_0$ and $u_1$ given. I know I can use a test form ...
7
votes
1answer
95 views

Another recurrence… $T(n)=\sqrt{2n} \cdot T(\sqrt{2n}) + \sqrt{n}$

I'm trying to solve the following recurrence : $$T(n)=\sqrt{2n}\cdot T(\sqrt{2n}) + \sqrt{n}$$ I've tried substituting $n$ for some other variables to transform the above to something easier without ...
7
votes
0answers
79 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
7
votes
0answers
220 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 !?
6
votes
5answers
199 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 $ab$ ...
6
votes
6answers
516 views

How to find the limit of this recurrence relation?

$a_n$ is a sequence where $a_1=0$ and $a_2=100$, and for $n \geq 2$: $$ a_{n+1}=a_n+\frac{a_n-1}{(n)^2-1} $$ I have a basic understanding of sequences. I wasn't sure how to deal with this recurrence ...
6
votes
2answers
527 views

Recursive square root problem [duplicate]

Give a precise meaning to evaluate the following: $$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$ Since I think it has a recursive structure (does it?), I reduce the equation to $$ ...
6
votes
4answers
139 views

closed form for $d(4)=2$, $d(n+1)=d(n)+n-1$?

I am helping a friend in his last year of high school with his math class. They are studying recurrences and proof by inference. One of the exercises was simply "How many diagonals does a regular ...
6
votes
3answers
381 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 ...
6
votes
5answers
148 views

$x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$ Proof

Prove $x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$. (Separate problems for $x_1 = 1$ and $x_1 = 27$.) EDIT: Took out bad algebra.
6
votes
3answers
87 views

Please solve this recurrence relation question for $8a_na_{n+1}-16a_{n+1}+2a_n+5=0$

Suppose $a_1=1$ and $$8a_na_{n+1}-16a_{n+1}+2a_n+5=0,\forall n\geq1,$$Please help to sort out the general form of $a_n$. Here are the first a few values of the series. Not sure if they are useful as ...
6
votes
3answers
224 views

Alternating Recurrence relation $a_n = b_{n-1} + 5$ and $b_n = na_{n-1}$

I am racking my brain on solving the relation where: $$a_n = b_{n-1} + 5$$ $$b_n = na_{n-1}$$ where $a_0$ = $b_0$ = 1 I am trying to find the closed form for $a_n$. I have tried to shifting $b_n = ...
6
votes
4answers
189 views

Recursive Sequence Tree Problem (Original Research in the Field of Comp. Sci)

This question appears also in http://cstheory.stackexchange.com/questions/17953/recursive-sequence-tree-problem-original-research-in-the-field-of-comp-sci. I was told that cross-posting in this ...
6
votes
3answers
324 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 ...