FoldUnfold Table of Contents Solving Linear Homogeneous Recurrence Relations with Generating Functions Solving Linear Homogeneous Recurrence Relations with Generating Functions Recall from the A Closed Form of the Fibonacci Sequence page that for the recursive definition of the Fibonacci numbers $f_n = f_{n-1} + f_{n-2}$ for $n \geq 3$ and with initial values $f_1 = […]

## Solving LHRRs with CCs and Repeated Roots of the Characteristic Equation

FoldUnfold Table of Contents Solving LHRRs with CCs and Repeated Roots of the Characteristic Equation Solving LHRRs with CCs and Repeated Roots of the Characteristic Equation Recall from the Repeated Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients page that if we have a linear homogeneous recurrence relation with constant coefficients of order $k$ […]

## Repeated Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients

FoldUnfold Table of Contents Repeated Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients Repeated Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients Consider the following linear homogeneous recurrence relation with constant coefficients of order $k$: (1) \begin{align} \quad f_n = a_1f_{n-1} + a_2f_{n-2} + … + a_kf_{n-k} \end{align} The characteristic equation for this […]

## Solving Linear Homogeneous Recurrence Relations with Constant Coefficients and Distinct Roots of the Characteristic Equation

FoldUnfold Table of Contents Solving Linear Homogeneous Recurrence Relations with Constant Coefficients and Distinct Roots of the Characteristic Equation Solving Linear Homogeneous Recurrence Relations with Constant Coefficients and Distinct Roots of the Characteristic Equation Recall from the Distinct Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients page that we saw if we had a […]

## Distinct Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients

FoldUnfold Table of Contents Distinct Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients Distinct Roots – Linear Homogeneous Recurrence Relations with Constant Coefficients Consider the following linear homogeneous recurrence relation with constant coefficients of order $k$ for $n \geq k$: (1) \begin{align} \quad f_n = a_1f_{n-1} + a_2f_{n-2} + … + a_kf_{n-k} \end{align} Which […]

## Vandermonde Matrices

FoldUnfold Table of Contents Vandermonde Matrices Vandermonde Matrices Consider the following linear homogeneous recurrence relation with constant coefficients $a_1, a_2, …, a_k$ of order $k$: (1) \begin{align} \quad f_n = a_1f_{n-1} + a_2f_{n-2} + … + a_kf_{n-k} \end{align} We will soon see that if the characteristic equation $x^k – a_1x^{k-1} – a_2x^{k-2} – … – […]

## Characteristic Equations of Linear Recurrence Relations

FoldUnfold Table of Contents Characteristic Equations of Linear Recurrence Relations Characteristic Equations of Linear Recurrence Relations Recall from the Linear Recurrence Relations page that a linear recurrence relation of order $k$ for the sequence $(f_n)_{n=0}^{\infty} = (f_0, f_1, f_2, …)$ is a linear recurrence of the form: (1) \begin{align} \quad f_n = a_1f_{n-1} + a_2f_{n-2} […]

## Linear Recurrence Relations

FoldUnfold Table of Contents Linear Recurrence Relations Linear Recurrence Relations Definition: A Linear Recurrence Relation of order $k$ for the sequence $(f_n)_{n=0}^{\infty} = (f_0, f_1, f_2, …)$ is a recurrence relation of the form $f_n = a_1f_{n-1} + a_2f_{n-2} + … + a_kf_{n-k} + b_n$ for $n \geq k$ where $a_1, a_2, …, a_{n-k}, b_n$ […]

## Hall’s Marriage Theorem

FoldUnfold Table of Contents Hall’s Marriage Theorem Hall’s Marriage Theorem Recall from the Systems of Distinct Representatives page that if $A$ is a finite set and $\mathcal A = (A_1, A_2, …, A_n)$ is a collection of subsets of $A$ then a system of distinct representatives of $\mathcal A$ is a collection of elements $x_1, […]

## Systems of Distinct Representatives

FoldUnfold Table of Contents Systems of Distinct Representatives Systems of Distinct Representatives Definition: Let $A$ be a finite set, and let $\mathcal A = ( A_1, A_2, …, A_n )$ be a collection of subsets of $A$. A System of Representatives of $\mathcal A$ is a collection of elements $x_1, x_2, …, x_n$ such that […]

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