N×3

N×3 denotes a rectangular structure with N rows and 3 columns, and by extension a board, matrix, or grid of that size. The notation follows the common m×n convention, where the first dimension is the number of rows and the second is the number of columns. The same board can be viewed as 3×N when rotated, with identical combinatorial properties.

In linear algebra, an N×3 matrix has N rows and 3 columns. Such matrices represent linear maps from a three-dimensional space to an N-dimensional space, and the set of all N×3 matrices over a field forms a vector space of dimension 3N. The rank of an N×3 matrix is at most 3, reflecting the number of columns.

In combinatorics and tiling problems, the N×3 grid is a common object of study. A classic question concerns tilings of a 3×N (equivalently N×3) rectangle with dominoes of size 2×1. The number of tilings is zero when N is odd. When N is even, the count, commonly denoted a(N), satisfies the recurrence a(N) = 4a(N−2) − a(N−4) with initial values a(0) = 1 and a(2) = 3. This sequence arises in dynamic programming formulations and has connections to linear recurrence relations and eigenvalue methods.

Beyond tilings, N×3 grids appear in computer graphics, data representation, and network modeling, where grid structure supports operations like traversal, partitioning, and matrix-based transformations. Problems on N×3 boards often illustrate fundamental techniques in combinatorics, linear algebra, and algorithm design, illustrating how a fixed column count interacts with variable row length to produce rich mathematical behavior.