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

8,270 questions

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Recursive system unique solution

General question: How does one identify if a system of recursive equations (a slightly more complicated system than the standard $Ax=b$) has a unique solution? Specific example: My system of two ... user avatar bissi

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Number of labelled triangular cactus graphs

I want to count triangular cactus graphs: I belive it can be done by creating recurrence realation, and then by analyzing its generating ... user avatar Tim Creig

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What is the general solution of the linear difference equation $y_{n+3}-4y_{n+2}+5y_{n+1}-2y_n=2^n$

What is the general solution to the linear difference equation: $$y_{n+3}-4y_{n+2}+5y_{n+1}-2y_n=2^n$$ I followed the instructions given in this video. And I got the following for the homogeneous part ... user avatar Yahya Fati

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Solving difference equation with Z transform reduces solution space?

I recently learned about the Z transform and I tried solving the equation: $$(1) : x[n]-x[n-1]-x[n-2]=\delta[n-1], x: \mathbb{Z} \rightarrow \mathbb{C}$$ which has the Fibonacci sequence (with $0$ at ... user avatar AggelosGkekas

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How should I find $a_n$ knowing that $a_n = a_{n-1} + a_{n-3}$

I tried using a quadratic formula by using the constants of the recursive formula. Then when I get the solutions of the quadratic function, I would insert the $x$ values gotten to $a_n = a_1 \cdot (... user avatar Wave S

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Second order linear recurrent sequence

Does anyone know the answer to this question regarding recurrence relations? To determine the general solution of the following recursive relation with the initial conditions. Write down the roots in ... user avatar bakerino

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Recurrence relations for even orthogonal polynomials

I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For ... user avatar miggle

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4votes2answers90views

Consider sequence of numbers $a_r,\;r\geq0\;$ with $a_0=1\;$ and $a_{r+1}^2=1+a_r\cdot a_{r+2}.\;$ Then which of the following is/are true?

Let $\alpha, \beta$ are roots of equation $x^2-a_1 x+1=0$ and consider sequence of numbers $a_r,\;r\geq0\;$ with $a_0=1\;$ and $a_{r+1}^2=1+a_r\cdot a_{r+2}.\;$ Then which of the following is/are true?... user avatar mathophile

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Solving recurrence relation $T(n) = T(n - 1) + n^2$ using mathematical induction.

I am trying to solve a recurrence using the induction principle (I am asking for help/confirmation about this solving method, so don't answer with a solution developed by iteration method or something ... user avatar Francesco

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How to represent this one-dimentional movement in math?

I'm struggling trying to convert this movement behavior into an equation. For future reference, this is as a result of searching for a solution for a previous question of mine. Thank you @eyeballfrog ... user avatar Jebyte

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How was this geometric product series simplified?

I feel very strange asking this but here is a problem I have from a textbook of mine. Simply put, I do not understand how (a_(t-1)*a_(t-2)…a_1) became the pi-product of a_i/a_o. As a result, I dont ... user avatar Vincent Dagene

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Convergence of a recursive equation defined by a concave, increasing function

Consider a function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}_{\geq 0}$ that is strictly increasing and concave, that is, $\frac{d f}{d x} > 0, \frac{d^2 f}{d x^2} < 0$, and suppose that $\lim_{x \... user avatar Yang

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Substitution method to solve recurrences.

Apparently, I didn't fully understood the substitution method to find upper/lower bounds of a recurrence relation. I know that this method have it's basis on the induction principle. For example: $$ T(... user avatar Francesco

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Solving recurrence T(n) = T(n - 1) + n^2 using substitution method.

I am trying to solve a recurrence using substitution method (I am asking for help/confirmation about this solving method, so don't answer with a solution developed by iteration method or something ... user avatar Francesco

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0votes2answers48views

How can I find an asymptotic solution to this recurrence?

How can I find an asymptotic solution to the recurrence $$T(n) = 4T(n/4) + 2T(n/2) + C$$ I replaced the $4T(n/4)$ with $4T(n/2)$ and used the master theorem to get an upper bound of $O(n^{\log_2 6})$ ... user avatar user236343

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