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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · fib(25)

Number-Theory · Sequences

Fib(25)

Àwọn ìṣàmúlò-ètò

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

Fi àwọn àgbèwọlé hàn

F_{25} = 75025

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Bii o ṣe le: Recurrences and generating functions

Each Fibonacci number is the sum of the two before it.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton Colouring and planar graphs