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

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9
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273 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 ...
7
votes
0answers
84 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
231 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
0answers
92 views

Mathematics of the Ice Bucket Challenge

I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ...
6
votes
0answers
91 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
6
votes
0answers
43 views

Recurrence relations with multiple roots of auxiliary equation

Suppose we have a homogeneous linear recurrence relation of the form $$u_{n+r}+a_1u_{n+r-1}+\dots +a_ru_n=0$$ The auxiliary equation is then $$f(x)=x^r+a_1x^{r-1}+\dots +a_r=0$$ It is well known that ...
6
votes
0answers
158 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
5
votes
0answers
86 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...
5
votes
0answers
35 views

Finding a recurrence relation for a sequence not containing $0$ in the digits. Also showing $\sum\limits_{k=1}^n \dfrac{1}{a_k} < 90$

Let $1,2,3,4,5,6,7,8,9,11,12,\ldots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that ...
5
votes
0answers
60 views

Right way to solve this reccurence relation?

I am not that good in mathematics , I have solved many simple relations. But I am not be able to solve the following. This is not a homework. I found it during analysis of a program. ...
5
votes
0answers
74 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 ...
5
votes
0answers
133 views

number of potential couples

A potential couple is a pair of a man and a woman that like each other (assume that 'like' is a symmetric relation). Given a group of $M$ men and $W$ women, I want to know how many different ...
5
votes
0answers
517 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
4
votes
0answers
52 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
4
votes
0answers
54 views

Transform recurrence relation

Is it possible to transform following recurrence relation $a_n=4a_{n-2}-a_{n-4}$, $a_0=1$, $a_1=0$, $a_2=3$, $a_3=0$ so that it will have nonnegative coefficients? Number of terms, of course, can be ...
4
votes
0answers
90 views

if $(n+1)c_{n}=nc_{n-1}+2nd_{n-1},d_{n}=2c_{n-1}+d_{n-1}$How prove $c_{n},d_{n}$ are integers

let sequences $c_{n},d_{n}$ such $$c_{0}=0,d_{0}=1$$ and for $n\ge 1$,have $$c_{n}=\dfrac{n}{n+1}c_{n-1}+\dfrac{2n}{n+1}d_{n-1}$$ $$d_{n}=2c_{n-1}+d_{n-1}$$ show that $c_{n},d_{n}$ are integers. my ...
4
votes
0answers
157 views

General overview about Recursion (free online texts)

I'm looking for free online texts about recursion. What I'm looking for formal definitions* of "all" (most of) the different types of recursion and from different points of views like Category ...
4
votes
0answers
177 views

A functional recursion problem..do you have any idea?

I have a problem which is related to algebra and polynomials. I would be very grateful if any of you could give a hand to solve it. Here is the problem: Consider the function $f_0(x) =x(1-x)$ and for ...
4
votes
0answers
167 views

Asymptotic behaviour of a two-dimensional recurrence relation

This problem comes out of a research in models of firm growth. The model is simple: A firm has two parameters which are its size (number of employees) and job vacancies. A firm of size $n$ will ...
4
votes
0answers
175 views

Is this a recurrence for the Mertens function plus 2?

If we define a symmetric array: $$T(1,1)=3,\; T(1,2)=2,\; T(2,1)=2$$ $$T(1,k)=\frac{-T(n,k-1)-\sum\limits_{i=2}^{k-1}T(i,k)}{k+1}+T(n,k-1)$$ $$ ...
3
votes
0answers
92 views

After how many steps can compositions of $x\mapsto x+1$ and $x\mapsto x^2+1$ produce the same result starting from $1$ and $3$?

This problem is from a Turkish contest: Let $P(x)=x+1$ and $Q(x)=x^2+1$. Consider all sequences $(x_k,y_k)$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k),Q(y_k))$ ...
3
votes
0answers
44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
3
votes
0answers
102 views

How to resolve this equation for f(n) without using f(n-1)

I have an equation related to some work I'm doing on Poisson distribution where I'm calculating a sequence of 100 values between a minimum and maximum value which is set by another formula. ...
3
votes
0answers
105 views

Lower Bound on fibonacci-like reccurence relation

Given the following recurrence: $T(n) := T(n-k) + T(n-1),\ n,k \in N$ $T(l) := 1,\ l \in \{0,...,k-1\}$ I need to find a lower bound for $T(n)$. (For $k = 2$, the recurrence is equal to the ...
3
votes
0answers
240 views

Two variable recurrence relations

I'm interested in solving the following type of problem... Starting with a recurrence relation in multiple variables, for example: $$ v_{i-1,j} + v_{i,j+1} + v_{i+1,j} + v_{i,j-1} = 4v_{i,j}$$ with ...
3
votes
0answers
58 views

An interesting partial recurrence equation

What is the closed form solution to the following partial recurrence relation? $$f(k,n) = \sum\limits_{t=0}^{m}f(k-t, n-1),$$ where $ m \geq 0$ is some fixed parameter. The boundary values are ...
3
votes
0answers
485 views

How to solve non-linear recurrence relation in general?

For linear recurrence, we can use generating function. So is there a general technique to solve non-linear recurrence or it depends on a specific sequence? For example, $$a_{n+1} = \dfrac{a_n(a_n - ...
3
votes
0answers
402 views

Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
3
votes
0answers
104 views

Perrin numbers in terms of the generalized hypergeometric function?

Given the roots of $x^3=x^2+1$, we have sequence A001609, $M(n) = x_1^n+x_2^n+x_3^n = \,_3F_2\left(\frac{-n}{3}, \frac{1-n}{3}, \frac{2-n}{3};\; \frac{1-n}{2}, \frac{2-n}{2};\; ...
3
votes
0answers
59 views

Properties preserved under the “reversal” of a recurrence equation

Consider the recurrence equation $u_n = f(u_{n-1},\ldots, u_{n-k})\;,$ defined for $n=0,1,2\ldots$. If this is a linear recurrence and the coefficient function on $u_{n-k}$ is nonzero in ...
2
votes
0answers
52 views

How to solve this finite-difference equation?

How to solve the following finite-differences equation: $$f(x) = f(x-1) + f(x-\sqrt{2}), \quad x\in [\sqrt{2}, +\infty) \,?$$ Let's say $f(x) = f_0(x)$ for $x \in [0, \sqrt{2})$ is a given function. ...
2
votes
0answers
39 views

Recurrence Relation with $n^{\frac{1}{2^k}}$ Square Root

I tried to solve this recurrence relation, like the following: $$ \\ T(n) = \begin{cases} T(\sqrt[2]{n}) + c, & \text{if }n \geq 2 \\ a, & \text{otherwise} \end{cases} \\\\ T(n) = ...
2
votes
0answers
73 views

Recurrence Relation Challenges

as i ask question and answered by some Clever people at this topic: Recurrence Relation Solving Problem i try to learn new thing with new question very similar to get familiar with recurrence ...
2
votes
0answers
31 views

Verify that this recurrence relation is in O(log n)

For the recurrence $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ is in $\Theta(lg n)$ My attempt at a solution (mostly just wanting to verify its correct). Lower Bound: $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ ...
2
votes
0answers
95 views

How to solve a ceiling expression or recurrence equation?

I am trying to express the following in term of n, $$\underbrace{\hbox{$ \left \lceil \frac{3}{2}.\left \lceil { \frac{3}{2}.\left \lceil { \ldots \left \lceil \frac{3}{2}.\left \lceil { ...
2
votes
0answers
135 views

Special map $f(x_n)=x_{{n+1} [\mod 4]}$

As mentioned in the headline I am considering the map $f(x_n)=x_{{n+1} [\mod 4]}$. It looks a little bit similar to the dyadic (bernoulli) map but instead of $\mod 1$ we have $\mod 4$ which means that ...
2
votes
0answers
51 views

(limit of) a linear second order recurrence relation with variable coefficients

I have the following recurrence relation: $(n + 1) a_{n + 2} = (w (n + 1) - c) a_{n + 1} - z (n + 1)*a_{n}$ that I would like to either solve, or to get the $n$ goes to Infinity limit of the ratio ...
2
votes
0answers
211 views

Induction proof of a recurrence relation

I have some trouble with an induction proof for the following problem. There is a vending machine that only takes coins of value 1 and 5 respectively. Let $S_n$ be the number of different ...
2
votes
0answers
33 views

Determinant of Stirling submatrix

Consider the table containing the Stirling numbers of the second kind $S\left( n,k\right) $. From this table construct a square matrix $M$ containing $N$ differents rows from the table ...
2
votes
0answers
88 views

How to solve this recurrence relation (related to discrete Fourier transform)?

I am having trouble with the following recurrence relation: $$c_{n+1} - c_{n-1} = 2\alpha \sin \frac{(2n-1)\pi}{N} c_n, \quad\forall n \in \mathbb{Z},$$ where $N$ is odd and the initial condition is ...
2
votes
0answers
59 views

Recurrence relation for polygamma reflection polynomials

In the reflection formula for the polygamma function $$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$ the right hand side is a polynomial ...
2
votes
0answers
94 views

Showing the relationship between Lucas and Fibonacci numbers with a recursive relationship

Given the equation $$L_{n+k}-(-1)^kL_{n-k}=5F_kF_n$$ Where $\{L\}$ is the set of Lucas numbers and $\{F\}$ is a Fibonacci sequence, $n$ and $k$ are both $>0$ and integers, how would one develop ...
2
votes
0answers
70 views

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
2
votes
0answers
51 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
2
votes
0answers
72 views

Solving recurrence relation of algorithm complexity?

Supposing I write an algorithm that results into this kind of recurrence relation $$\left\{ \begin{array}{ll} T(0)=T(1)=1 \\ T(n)=T\left(\lfloor n/2 \rfloor \right)+T\left(\lceil n/2 ...
2
votes
0answers
118 views

Recursive Sequence from Finite Sequences

I'm searching for the name of these kind of sequences: Suppose you start off with a finite sequence containing one term: S0 = {3} To get the next sequence you ...
2
votes
0answers
99 views

Is there a closed form for this recurrence?

Given $$ E_{n,k} = \begin{cases} 0 & \text{ if } n \leq k \\ n & \text{ if } k = 0 \\ \sum_{i=0}^{n-1} \dfrac{1}{n} \cdot E_{i,k-1} & \text{ otherwise } \end{cases} $$ I wonder is there ...
2
votes
0answers
66 views

How much information do I gain from each modular inequality?

Problem details: Let $p = 2^{48}$, $s\leftarrow\{0,1\}^{48}$, and $a,b$ known constants. Furthermore let $f(x) = a x + b \pmod{p}$ and let the value $r_k$ be defined by the first-order recurrence ...
2
votes
0answers
54 views

how to compute $\lim_{n\to\infty}\ln(f(n))/\ln(n)$ for this given $f(n)$?

Let $c=f(0)$ be a real number and $n,m$ positive integers. Let $P_k(n)$ be integer polynomials of $n$. Let $f(n)$ be defined by the recursion $\left[\sum_{k=0}^{m} P_k(n)f(n+k)\right]+P_{-1}(n)=0$. ...
2
votes
0answers
548 views

Probability of tossing a biased coin without having k heads consecutively in a row

I got asked by a friend this question; I have a coin, the probability of receiving a head by tossing is $p$ and tail $1-p$. I have to toss it $n$ times without getting $k$ heads in a row. What is the ...