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‎src/algebra/fibonacci-numbers.md

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To decode a code word, first remove the final $1$. Then, if the $i$-th bit is set (indexing from 0 from the leftmost to the rightmost bit), sum $F_{i+2}$ to the number.
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## Formulas for the n-th Fibonacci number
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## Formulas for the $n^{\text{th}}$ Fibonacci number { data-toc-label='Formula for the <script type="math/tex">n^{\text{th}}</script> Fibonacci number' }
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The $n$-th Fibonacci number can be easily found in $O(n)$ by computing the numbers one by one up to $n$. However, there are also faster ways, as we will see.
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‎src/algebra/sieve-of-eratosthenes.md

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# Sieve of Eratosthenes
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Sieve of Eratosthenes is an algorithm for finding all the prime numbers in a segment $[1;n]$ using $O(n \log \log n)$ operations.

‎src/combinatorics/binomial-coefficients.md

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# Binomial Coefficients
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Binomial coefficients $\binom n k$ are the number of ways to select a set of $k$ elements from $n$ different elements without taking into account the order of arrangement of these elements (i.e., the number of unordered sets).

‎src/combinatorics/bishops-on-chessboard.md

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# Placing Bishops on a Chessboard
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Find the number of ways to place $K$ bishops on an $N \times N$ chessboard so that no two bishops attack each other.

‎src/combinatorics/bracket_sequences.md

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# Balanced bracket sequences
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A **balanced bracket sequence** is a string consisting of only brackets, such that this sequence, when inserted certain numbers and mathematical operations, gives a valid mathematical expression.

‎src/combinatorics/burnside.md

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# Burnside's lemma / Pólya enumeration theorem
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## Burnside's lemma

‎src/combinatorics/catalan-numbers.md

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# Catalan Numbers
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Catalan numbers is a number sequence, which is found useful in a number of combinatorial problems, often involving recursively-defined objects.
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‎src/combinatorics/counting_labeled_graphs.md

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# Counting labeled graphs
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## Labeled graphs

‎src/combinatorics/generating_combinations.md

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# Generating all $K$-combinations
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In this article we will discuss the problem of generating all $K$-combinations.

‎src/combinatorics/inclusion-exclusion.md

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# The Inclusion-Exclusion Principle
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The inclusion-exclusion principle is an important combinatorial way to compute the size of a set or the probability of complex events. It relates the sizes of individual sets with their union.

‎src/combinatorics/stars_and_bars.md

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# Stars and bars
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Stars and bars is a mathematical technique for solving certain combinatorial problems.

‎src/contrib.md

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# How to Contribute
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## General information

‎src/data_structures/deleting_in_log_n.md

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# Deleting from a data structure in $O(T(n)\log n)$
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Suppose you have a data structure which allows adding elements in **true** $O(T(n))$.

‎src/data_structures/disjoint_set_union.md

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# Disjoint Set Union
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‎src/data_structures/fenwick.md

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# Fenwick Tree
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‎src/data_structures/randomized_heap.md

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# Randomized Heap
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A randomized heap is a heap that, through using randomization, allows to perform all operations in expected logarithmic time.

‎src/data_structures/segment_tree.md

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# Segment Tree
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‎src/data_structures/sparse-table.md

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# Sparse Table
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‎src/data_structures/sqrt-tree.md

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# Sqrt Tree
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‎src/data_structures/sqrt_decomposition.md

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‎src/data_structures/stack_queue_modification.md

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‎src/data_structures/treap.md

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# Treap (Cartesian tree)
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Treap is a data structure which combines binary tree and binary heap (hence the name: tree + heap $\Rightarrow$ Treap).

‎src/demo-article.md

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# Demo Article
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‎src/dynamic_programming/divide-and-conquer-dp.md

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# Divide and Conquer DP
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‎src/dynamic_programming/profile-dynamics.md

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# Dynamic Programming on Broken Profile. Problem "Parquet"
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‎src/dynamic_programming/zero_matrix.md

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# Finding the largest zero submatrix
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‎src/game_theory/games_on_graphs.md

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# Games on arbitrary graphs
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Let a game be played by two players on an arbitrary graph $G$.

‎src/game_theory/sprague-grundy-nim.md

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# Sprague-Grundy theorem. Nim
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## Introduction

‎src/geometry/area-of-simple-polygon.md

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‎src/geometry/basic-geometry.md

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# Basic Geometry
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‎src/geometry/check-segments-intersection.md

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# Check if two segments intersect
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You are given two segments $(a, b)$ and $(c, d)$.

‎src/geometry/circle-circle-intersection.md

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# Circle-Circle Intersection
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‎src/geometry/circle-line-intersection.md

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‎src/geometry/convex-hull.md

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# Convex Hull construction
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In this article we will discuss the problem of constructing a convex hull from a set of points.

‎src/geometry/convex_hull_trick.md

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# Convex hull trick and Li Chao tree
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‎src/geometry/delaunay.md

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# Delaunay triangulation and Voronoi diagram
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Consider a set $\\{p_i\\}$ of points on the plane.

‎src/geometry/halfplane-intersection.md

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‎src/geometry/intersecting_segments.md

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‎src/geometry/lattice-points.md

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# lattice points inside non-lattice polygon
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For lattice polygons there is Pick's formula to enumerate the lattice points inside the polygon.
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