3xN

3xN refers to the problem of tiling a 3-by-N rectangular grid with 2-by-1 dominoes (placed either horizontally or vertically). This classical combinatorial problem asks for the number of distinct tilings for each integer N, and is often studied using recurrence relations and transfer-matrix methods.

A key property is that tiling is only possible when N is even. Since each domino covers two squares, the area of the board is 3N, which is even only if N is even. Consequently, the number of tilings T(N) is zero for odd N, while it is positive for even N.

The standard recurrence for the number of domino tilings of a 3-by-N rectangle is: T(0) = 1, T(2) = 3, and for even n ≥ 4, T(n) = 4*T(n-2) − T(n-4). This relation arises from a decomposition of the board into smaller tiling problems and accounts for both straightforward and symmetric tilings.

Examples of small values: T(0) = 1, T(1) = 0, T(2) = 3, T(3) = 0, T(4) = 11, T(6) = 41, T(8) = 153. These illustrate the rapid growth and the zero values for odd N.

A convenient way to study the sequence is to define S(k) = T(2k). Then S(0) = 1, S(1) = 3, and S(k) = 4*S(k-1) − S(k-2) for k ≥ 2. This yields a closed-form expression in terms of the roots of x^2 − 4x + 1 = 0, namely 2 ± √3, giving S(k) as a linear combination of (2+√3)^k and (2−√3)^k.

See also domino tiling, transfer-matrix method, and Fibonacci-like sequences.