Questions tagged [recurrence-relations]

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

A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values: once one or more initial terms are given, each further term of the sequence or array is defined as a function of the preceding terms.

Simple examples include the geometric sequence $a_{n}=r a_{n-1}$, which has the closed-form $a_{n}=r^n a_0$, the aforementioned Fibonacci sequence with initial conditions $f_0=0,f_1=1$ and recurrence $f_{n+2}=f_{n+1}+f_n$, and series: the sequence $S_n =\sum_{k=1}^{n} a_k$ can be written as $S_n= S_{n-1}+a_n$.

The term order is often used to describe the number of prior terms used to calculate the next one; for instance, the Fibonacci sequence is of order 2.

See the Wikipedia page for more information.

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I want to know the limits and convergence/divergence of ${a_{n+1}} = \frac{(a_n^2 + a_n + 2)}{6}$

I`ve the following iteratively defined sequence: $${a_{n+1}} = \frac{(a_n^2 + a_n + 2)}{6}$$ $${a_1=0}\;\text{for}\;{n\ge1}$$ I want to know the limits and if it is convergent or divergent, but I don't know how to handle two "$a_n$'s" in the…
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Could you please check if this substitution is right so far?

The question: Use resubstitution to solve the following recurrence equation: $$T(n) = 2T(n-1) + n;\; n \ge2\text{ and }T(1) = 1.$$ So far I have this: $$\begin{align}T(n) &= 2T(n-1) + n\\ &= 2(2T(n-2) + (n-1)) + n\\ &= 4T(n-2) + 3n -2\\ &= 2(4T(n-3)…
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Where to start? Sequence of functions.

I've been playing around with composing translations with the modulus function. Let $f_0(x) = |x|$, and $f_{n+1}(x) = \left| f_n(x)-\frac{1}{n+1}\right|$, where $n$ is a positive integer. I would like to understand $\displaystyle{\lim_{n \to…
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Integer solutions to $\sqrt{ax^2+1}$ and the unusually large solutions

While trying to simplify some triangle graphing I came across this equation $\sqrt{ax^2+1}$ looking for integer solutions. Trivially $x=0$ is true for any $a$. However the non-trivial solutions have a nice recurrent relation. For example,…
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How to solve second degree recurrence relation?

For first degree recurence relation it is as simple as $f(n)=a^n\cdot f(0)+b\dfrac{a^n-1}{a-1}$. But how do you solve second degree? For example $$f(n)=\begin{cases} 1,&\text{for }n=1\\ 2,&\text{for }n=2\\ -3f(n-1)+4f(n-2),&\text{for…
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Help in solving linear recurrence relation

I need to solve the following recurrence relation: $a_{n+2} + 2a_{n+1} + a_n = 1 + n$ My solution: Associated homogeneous recurrence relation is: $a_{n+2} + 2a_{n+1} + a_n = 0$ Characteristic equation: $r^2 + 2r + 1 = 0$ Solving the characteristic…
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Bags, probability and recurrence

I need help on this one: There are two bags, each of them contain $n$ balls. The first bag contains only white balls. The second bag contains only black balls. Pick one black ball and input it to first bag. Then choose one ball from first bag…
M. Red
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Common theory for Linear equations, linear ODEs and linear recurrence relations

In Linear equations, linear ODEs and linear recurrence relations, when solving homogeneous equations, there are a subspace of solutions, and when solving inhomogeneous equations, a particular inhomogeneous solution plus the subspace of homogeneous…
Tim
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Recurrence for the number of ternary strings of length $n$ that contain either two consecutive $0$s or two consecutive $1$s

I attempted this problem and this is what I have so far: First, I considered the possible "cases". If the string starts with $00$ or $11$, then the rest can be anything so there are $2\cdot 3^{n-2}$ such strings. If the string starts with $2$,…
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Sylvester's sequence, and a similar sum

Sylvester's Sequence is defined as $$e_{n+1}=e_n^2-e_n+1,\space\space e_0=2$$ and it has the interesting property that $$\sum_{k=0}^\infty \frac{1}{e_k}=1$$ despite the fact that there is no pretty closed-form explicit formula for $e_n$. This makes…
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How does this numerical method of root approximation work?

We can often use iteration to find an approximation for the root of the equation $f(x) = 0$. For instance, consider $x^2 - 4x + 1 = 0$. This is equivalent to $x = 4 - \frac 1x$. Now we use the iteration formula $$x_{n+1} = 4 - \frac{1}{x_n}$$…
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Proof of this pattern in the root 2 algorithm.

I'm looking this recursive formula to approximate $\sqrt{2}$ \begin{align*} x_1 &= 2 \\ x_{n+1} &= \frac{1}{2}\left(x_n + \frac{2}{x_n}\right) \end{align*} I noticed an easier way to approximate by hand would be to think of each $x_k$ as a rational…
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Integer sequences uniquely defined by functional relations involving iteration

Here is a generalized version of a high school olympiad problem I saw a while ago (1985 Tournament of the Towns, to be specific): Find a strictly increasing function $f:\mathbb{N}\to\mathbb{N}$ (where $\mathbb{N}$ does not include $0$ here) that…
Alexander Burstein
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Muller's recurrence: limit

In Kahan's account of the Muller's recurrence (p.16): $x_{n+1} = E(x_n, x_{n-1})$ for the function $$ E(y, z) = 108 - \frac{815-\frac{1500}{z}}{y}, $$ he uses the characteristic polynomial for the recurrence to deduce the closed form: $$ x_n =…
lauren96
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Solving recursion $f(n+1)=\frac n{n+2}f(n)$

How can I find an expression for $$f(n), n \in \mathbb{N}$$ if $$f(n+1)=\frac{n}{n+2}\cdot f(n)$$ and $$f(1) = 999$$ I'm used to solve simple homogeneous recursion relations by the chacarcteristic equation, but it seems impossible in this situation.…
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