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The Encyclopedia of Integer Sequences
Humans are pattern-seeking story-telling animals, and we are quite adept at telling stories about patterns, whether they exist or not. — Michael Shermer
The Online Encyclopedia of Integer Sequences (OEIS) is exactly what it sounds like. A database of different sequences of integers is useful for researchers to identify known integer sequences, find formulas, and discover connections between different areas of mathematics.
There are sequences that surprisingly transcend different areas of mathematics and other disciplines. The Catalan numbers (A000108) 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862. The Catalan numbers solve the following problems:
The number of valid combinations of n pairs of parentheses.
The number of distinct binary trees that can be formed with n nodes.
The number of ways to divide a convex polygon with n + 2 sides into triangles by drawing non-intersecting diagonals.
The number of monotonic paths along the edges of a grid that do not cross above the diagonal.
The number of ways that 2n people sitting around a table can pair up for handshakes without any arms crossing.
The number of ways a stack can be sorted by a series of push and pop operations.
The number of ways to fully parenthesize a product of n matrices.