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Recurrences and generating functions
Solving recurrence relations, and encoding a sequence as a power series.
A recurrence defines aₙ from earlier terms; a generating function Σ aₙ xⁿ packs the whole sequence into one function whose algebra does the counting. Picture it: the Fibonacci numbers growing by the golden ratio each step. Think it: linear recurrences with constant coefficients are solved exactly like constant-coefficient differential equations, a characteristic polynomial.
Worked example · fib(25)
Step by step
- F_0 = 0,; F_1 = 1
Each Fibonacci number is the sum of the two before it.
- F_{2} = F_{1} + F_{0} = 1
- F_{3} = F_{2} + F_{1} = 2
- F_{4} = F_{3} + F_{2} = 3
- F_{5} = F_{4} + F_{3} = 5
- F_{6} = F_{5} + F_{4} = 8
- F_{7} = F_{6} + F_{5} = 13
- F_{8} = F_{7} + F_{6} = 21
- \vdots
- F_{24} = F_{23} + F_{22} = 46368
- F_{25} = F_{24} + F_{23} = 75025
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How to: Recurrences and generating functions
- Each Fibonacci number is the sum of the two before it.
Questions people ask
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
More in Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs