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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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title: Sieve of Eratosthenes
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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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title: Placing Bishops on a Chessboard
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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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title: Burnside's lemma / Pólya enumeration theorem
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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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title: Catalan Numbers and its Applications
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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.

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