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Mathematics
Combinatorics
1. Introduction to Combinatorics
2. Fundamental Counting Principles
3. Permutations
4. Combinations
5. The Binomial Theorem
6. Advanced Counting Techniques
7. Recurrence Relations
8. Generating Functions
9. Special Counting Numbers and Sequences
10. Graph Theory and Combinatorics
11. Design Theory
12. Probabilistic Combinatorics
13. Algebraic Combinatorics
14. Extremal Combinatorics
15. Applications and Advanced Topics
7.
Recurrence Relations
7.1.
Introduction to Recurrence Relations
7.1.1.
Definition and Examples
7.1.2.
Initial Conditions
7.1.3.
Order of Recurrence Relations
7.1.4.
Modeling Problems with Recurrences
7.2.
Classic Examples
7.2.1.
Tower of Hanoi
7.2.2.
Fibonacci Numbers
7.2.3.
Catalan Numbers
7.2.4.
Tribonacci Numbers
7.3.
Linear Homogeneous Recurrence Relations
7.3.1.
Definition and Structure
7.3.2.
First-Order Relations
7.3.3.
Second-Order Relations
7.3.4.
The Characteristic Equation
7.3.5.
Distinct Roots Case
7.3.6.
Repeated Roots Case
7.3.7.
Complex Roots Case
7.3.8.
General Solutions
7.4.
Linear Non-Homogeneous Recurrence Relations
7.4.1.
Definition and Structure
7.4.2.
Homogeneous and Particular Solutions
7.4.3.
Method of Undetermined Coefficients
7.4.4.
Variation of Parameters
7.4.5.
Superposition Principle
7.5.
Advanced Techniques
7.5.1.
Substitution Methods
7.5.2.
Generating Function Approach
7.5.3.
Matrix Methods
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8. Generating Functions