"In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms."
A sequence or function defined in terms of its previous term(s). Used to model dynamic processes and solve problems in combinatorics, graph theory, and computer science.
Introduction to Recurrence Relations: This topic covers the definition of recurrence relations and their importance in combinatorics.
First-Order Linear Recurrence Relations: This topic deals with the first-order linear recurrence relations and how to find the solutions to such relations.
Second-Order Linear Recurrence Relations: This topic covers the second-order linear recurrence relations and how to find the solutions to such relations.
Non-Linear Recurrence Relations: This topic focuses on the non-linear recurrence relations and different methods to solve them.
Homogeneous Recurrence Relations: This topic deals with the homogeneous recurrence relations and how to find the solutions to such relations.
Inhomogeneous Recurrence Relations: This topic covers the inhomogeneous recurrence relations and different methods to find the solutions to such relations.
Generating Functions: This topic deals with the concept of generating functions in recurrence relations and how to use them to find closed-form solutions.
Dividing Recurrence Relations: This topic focuses on the concept of dividing recurrence relations and how to use it to find closed-form solutions.
Master Theorem: This topic covers the Master Theorem in recurrence relations and how to use it to find closed-form solutions.
Special Sequences and Recurrence Relations: This topic focuses on special sequences such as the Fibonacci sequence and the solutions to the corresponding recurrence relations.
Applications of Recurrence Relations: This topic covers the various applications of recurrence relations in combinatorics, such as in counting problems and probability calculations.
Divide and Conquer Recurrence Relations: These are recurrence relations that arise in divide and conquer algorithms, where the sequence is split into smaller sub-sequences. For example: a_n = 2a_n/2 + n.
Recursive Sequences: These are recurrence relations that define a sequence in terms of itself. For example: a_n = a_n-1 + n.
Fibonacci-like Sequences: These are recurrence relations that define sequences similar to the Fibonacci sequence, where each term is the sum of the two previous terms. For example: a_n = a_n-1 + a_n-2.
Geometric Sequences: These are recurrence relations where the terms of the sequence follow a geometric progression. For example: a_n = 2a_n-1.
Convergent Sequences: These are recurrence relations where the terms of the sequence converge to a limit or approach a constant value. For example: a_n = 1/n.
Oscillating Sequences: These are recurrence relations where the terms of the sequence oscillate or alternate between positive and negative values. For example: a_n = (-1)^n/n.
"This number k is called the order of the relation."
"The rest of the sequence can be calculated by repeatedly applying the equation."
"In linear recurrences, the nth term is equated to a linear function of the k previous terms."
"A famous example is the recurrence for the Fibonacci numbers."
"where the order k is two and the linear function merely adds the two previous terms."
"For these recurrences, one can express the general term of the sequence as a closed-form expression of n."
"Linear recurrences with polynomial coefficients depending on n are also important."
"Because many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation."
"Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n."
"Yes, the concept of a recurrence relation can be extended to multidimensional arrays."
"That is, indexed families that are indexed by tuples of natural numbers."
"For these recurrences, one can express the general term of the sequence as a closed-form expression of n."
"Many common elementary and special functions have a Taylor series whose coefficients satisfy such a recurrence relation."
"Obtaining a closed-form solution: a non-recursive function of n."
"This number k is called the order of the relation."
"The rest of the sequence can be calculated by repeatedly applying the equation."
"The nth term is equated to a linear function of the k previous terms."
"One can express the general term of the sequence as a closed-form expression of n."
"That is, indexed families that are indexed by tuples of natural numbers."