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

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12
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302 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
66 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 ...
7
votes
0answers
91 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 ...
7
votes
0answers
85 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
246 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
94 views

Minimizing beginning terms of a Fibonacci-like sequence

For $ a, b \in \mathbb{Z}^+ $ such that $ a = b $ or $ 2a < b $, let $ f_{a, b} : \mathbb{Z}^+ \to \mathbb{Z}^+ $ be a sequence that satisfies the following three criteria: $$ f(1) = a, \\ f(2) = ...
6
votes
0answers
122 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
97 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
166 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
93 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
38 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
64 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
141 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
539 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
31 views

Solving system of nonlinear difference equations

This is perhaps a trivial question (I’m an economist), but as a non-matematician that might elude me. I have a (two variable) system of nonlinear difference equations (similar to Riccati type ...
4
votes
0answers
82 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
94 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
160 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
153 views

The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), ...
4
votes
0answers
179 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
170 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
177 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
37 views

Relation between 2 recurrence equations.

I am stuck with the following recurrence relations (actually asked in a programming question).Given the following relationships, is it possible to find the index of the occurrence of any given number ...
3
votes
0answers
30 views

How to work with a recursive function with 2 recursive instances?

In class, we figured out how to find the closed form of a recursive definition through the "basic 5 steps method". Example function T(n): If n = 1, T(1) = 1 If n > 1, T(n) = T(n-1)+1 Step 1: ...
3
votes
0answers
53 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
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
96 views

What is this bifurcation?

I have a discrete dynamical System $x_{n+1}=f(x_{n},x_{n-1},x_{n-2},x_{n-3},x_{n-4},\lambda)$ with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger ...
3
votes
0answers
80 views

About linear recurrence sequences

Let $\{a_n\}_{n=0}^\infty$,$\{b_n\}_{n=0}^\infty$,$\{c_n\}_{n=0}^\infty$ be three complex sequences and satisfy \begin{eqnarray*} &&\sum_{k=0}^2\alpha_ka_{n+k}=0,\\ ...
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
110 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
248 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
60 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
634 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
454 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
105 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
60 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
18 views

Deriving a recurrence equation and apply Master's Thrm to it

So for a function such as: function Hi(n) if n > 1 then for j ← 1 to n do print(”Hi”) Hi(n/2) Hi(n/2) Hi(n/2) It's very easy to eyeball this and get a ...
2
votes
0answers
29 views

Solve the recurrence $T(n)=aT(n-1)+bn$

I have to solve the following recurrence, given $T(1)=1$, $$T(n)=aT(n-1)+bn$$ I have done the following: $$T(n)=aT(n-1)+bn \\ =a^2T(n-2)+ab(n-1)+bn \\ =a^3T(n-3)+a^2b(n-2)+ab(n-1)+bn \\ = \dots \\ ...
2
votes
0answers
33 views

What are the main similarities between difference equations and differential equations, specifically in methods for solving them?

I know that that difference equations can be used to represent discrete dynamical systems and differential equations can be used to represent continuous dynamical systems. Therefore both are ...
2
votes
0answers
67 views

A similar, but hopefully easier problem than Gilbreath's conjecture

Gilbreath's conjecture says that for every positive integer $n$, if we write out the first $n$ primes $2,3,5,7,11,13,\ldots,p_n$ take the differences between consecutive terms ...
2
votes
0answers
41 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
2
votes
0answers
30 views

How to solve a first order inhomogenous recurrence relation?

I have a recurrence relation for a fund that starts a 50 million, 6 % interest every year, and an outtake of 2 million/year. How to find out a solution for what funds exists after n years? ...
2
votes
0answers
28 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
2
votes
0answers
59 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
45 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
63 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
2
votes
0answers
37 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
99 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
137 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 ...