2×n

2×n is a common mathematical notation that can denote a two-by-n rectangle, a two-by-n matrix, or the product of 2 and n depending on context. In geometry and combinatorics, a 2×n rectangle consists of two rows and n columns of unit squares, for a total area of 2n. A classical problem concerns tiling a 2×n rectangle with 2×1 dominoes. The number of tilings, a_n, satisfies a_0 = 1, a_1 = 1, and a_n = a_{n-1} + a_{n-2} for n ≥ 2; hence a_n equals the (n+1)th Fibonacci number. This recurrence is a standard example in teaching dynamic programming and recurrence relations.

In linear algebra, a 2×n matrix is an array with two rows and n columns. Its elements come from a specified field, ring, or set. The matrix has rank at most 2, and its transpose is an n×2 matrix. Determinants are defined only for square matrices; therefore a 2×n matrix with n ≠ 2 does not have a determinant. Row and column operations can be used to reduce it to a form that reveals independent rows or columns, and the row space has dimension at most 2.

In arithmetic, 2×n denotes the product 2n. This value is even for any integer n and is linear in n. It satisfies standard properties such as 2(n+m) = 2n + 2m and (2n)m = 2(nm). When n is variable, 2×n is often used to describe even multiples and to express sequences of even numbers.

Overall, the meaning of 2×n is context dependent, with common usages in geometry, linear algebra, and arithmetic.