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-4
votes
0answers
63 views

Recurrence relation practice problem that I can't figure out [closed]

thanks for taking the time to look at this problem. I'm trying to prepare for a test on Monday by doing some extra odd numbered problems from my textbook. I'm having a lot of trouble trying to solve ...
7
votes
1answer
378 views

Hamming weight of Fibonacci numbers

The Hamming weight $w(n)$ is the number of 1s in $n$ when written in binary. Is there some effective bound on Fibonacci numbers $F_n$ with $w(F_n)\le x$ for a given $x$? Clearly only $F_0=0$ has ...
1
vote
1answer
163 views

Infinitely many sufficiently large powers in linear recurrences

Edit Aaron solved the original question with the fourth order $$ a(n)=n2^n+\frac{(-1)^n-1^n}{2} $$ trying to make the question harder. Let $a(n)$ be a linear recurrence with constant coefficients, of ...
9
votes
3answers
487 views

Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
1
vote
0answers
83 views

Bound a sum of a serie defined by a recursive integer function

I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as \begin{equation} f(n)=\lceil \log(f(n-1)) \rceil +f(n-1) \end{equation} where $f(1)=F\in \mathbb{N}$, and ...
1
vote
1answer
142 views

Does there exist a recurrence relation which cannot be written in a closed form? [closed]

I'm looking for an example of a recurrence relation where it provably does not have a closed form. Does such a thing exist? For example, I believe $S(n, k)$, or the Stirling numbers of the second ...
0
votes
1answer
95 views

Solving a two dimensional non-homogenous linear recurrence

How one can solve the following recurrence: \begin{align} X[i,0] &=0 \quad \forall i =1,\ldots, m\\ X[m,n] &= a_n X[m,n-1]+b_n \sum_{i=k_m}^{m-1}X[i,i] +c_n \end{align} Where $a_i\ge 1 ,~ 0 ...
5
votes
1answer
190 views

Asymptotic growth of recurrence relation $x_n=\min\limits_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2)$

Suppose $a>0$ . Define $$x_n=\min_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2) \text{ with } x_1=0.$$ Can we find $r>0$ such that there exists two positive constant $c_1,c_2$ such that ...
23
votes
4answers
929 views

Multiplying by irrational numbers in combinatorial problems

This is getting no attention on stackexchange. Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$. It had escaped my attention until last week, ...
-1
votes
1answer
90 views

Recursion equation convergence to sqrt [closed]

I'm new here. I have a recursion equation $$ a[n] = \frac{n \cdot a[n-1]}{n+a[n-1]} + k $$ $$ a[1] = 1 $$ $$ k \in \mathbb{R}, k > 0 $$ and from computing and ploting graphs I found that it can ...
5
votes
1answer
788 views

Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
5
votes
0answers
110 views

Non-crystallographic cluster algebras

Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed ...
0
votes
0answers
67 views

Approximate closed-form solution for a recurrence

Find an (approximate) closed-form solution for $S(m, b)$. $$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad + \sum_{i=\lfloor (e-1)/2\rfloor+1}^{\min(b,e)}{e\choose ...
1
vote
1answer
119 views

Closed form solution to an iterative equation.

$f(n+1) = f(n) + f(n)^{a}$ where $a \in (0,1)$ and $n \ge 1$ with $f(1) = m$. If $a=0$, we see $f(n) = m + n - 1$ and if $a=1$, we see $f(n) = 2^{n-1}m$. So the recursion seems to interpolate ...
0
votes
1answer
183 views

Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$. Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise Let ...
6
votes
3answers
500 views

References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
2
votes
0answers
134 views

Master theorem for probabilistically inspired recurrences

Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis? $f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j,k-1) + p_4 ...
4
votes
3answers
615 views

Linear Recurrence Relations in 2 Variables with Variable Coefficients

Consider the following recurrence relation: $$-2a_{n,m} +a_{n-1,m}+a_{n,m-1}=0,$$ where $a_{n,m} \in \mathbb{C}.$I would like a purely combinatorial way to understand the subspace of solutions to this ...
1
vote
2answers
287 views

Methods for solving two variable recurrence

I have a recurrence $$f(i,j) = 1+ \frac{N-i}{N} f(i,j-1) + \frac{i}{N} f(i-1, j)$$ $$f(i,0) = 0$$ $$f(0,j) = j$$ I would like to compute $f(N,M)$ in terms of N and M. The system is defined for ...
3
votes
1answer
170 views

Books request on nonlinear recurrence relations.

Hi, do you have some sort of a bibliography on advanced techniques in recurrence equations, such as nonlinear ones and others? As I see it recurrence equations are quite similar to differential ...
1
vote
2answers
94 views

A recursive Double sequence related to uniform Cardinal B-spline

Given a sequence $A_n(k)$ defined as follows: $A_0(0)=1$, $A_0(k)=0$ for all nonzero integers $k$ and $$A_{n}(k)=(n+1-k)^2A_{n-1}(k-1)+2(n(n+1)-k^2)A_{n-1}(k)+(n+1+k)^2A_{n-1}(k+1)$$ for all positive ...
1
vote
0answers
94 views

Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...
2
votes
0answers
83 views

Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence $$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), ...
0
votes
0answers
375 views

Solving a linear recurrence relation with variable coefficients.

I have the following recurrence relation: \begin{equation} A[n]=f_A[n-1] A[n-1] + f_B[n-1]B[n-1], \\ B[n]=g_A[n-1] A[n-1] + g_B[n-1]B[n-1], \end{equation} where ...
0
votes
0answers
96 views

Recursive relation using successor function

What is the recursive relation for H(m)=2^(m^2) using successor function recursive relation for multiplication: mult(x,0)=0; mult(x,S(y))=add(x,mult(x,y)) recursive relation for addition: add(x,0)=x; ...
1
vote
0answers
149 views

Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets. He proved in Theorem 8.8 that in each finite partition of ...
6
votes
5answers
451 views

Uniqueness of values in recurrence relations

Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows: $a=4k-2,$ $y_0 = 1,$ $y_1 = a + 1,y_n = ay_{n-1} - y_{n-2}$ $b = 4k + 2,$ $ x_0 = 1,$ $x_1 = b - 1,$ $x_n = ...
14
votes
3answers
499 views

A question on the Laurent phenomenon

This question is motivated by my answer to 109955. It gives a recurrence relation satisfied by a function $P(n)$ whose terms a priori are rational functions (of three variables) with complicated ...
2
votes
2answers
595 views

How to solve a specific multivariate recurrence relation (or general ones)

How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$. ...
2
votes
1answer
534 views

Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$: $$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - ...
1
vote
1answer
185 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
votes
4answers
1k views

Recurrence T(N)=T(N/LOGN)+1 [closed]

I'm stuck on the following reccurrence : T(N)=T(N/LOGN)+1,IF N>2 T(N)=0,if 0<=N<=2 I need a function T(N) for all N>0 Is there some method for solving ...