có chỉ là một cách để gạch hình chữ nhật 2 x 0 với domino, cụ thể là để sử dụng không có cờ domino, vì vậy Đến = 1. (Chiến lợi phẩm T mô hình đơn giản, = n đúng khi n = 1, 2 và 3, nhưng mô hình mà có thể phải chịu số phận anyway, | DOMINO THEORY AND CHANGE 307 there s just one way to tile a 2 X 0 rectangle with dominoes namely to use no dominoes therefore To 1. This spoils the simple pattern Tn n that holds when n 1 2 and 3 but that pattern was probably doomed anyway since To wants to be 1 according to the logic of the situation. A proper understanding of the null case turns out to be useful whenever we want to solve an enumeration problem. Let s look at one more small case n 4. There are two possibilities for tiling the left edge of the rectangle-we put either a vertical domino or two horizontal dominoes there. If we choose a vertical one the partial solution is fl I and the remaining 2x3 rectangle can be covered in Tj ways. If we choose two horizontals the partial solution FTI can be completed in T2 ways. Thus T4 Tj T2 5. The five tilings are mi ÍTR El HU and FB. We now know the first five values of Tn n Tn 0 12 3 4 112 3 5 These look suspiciously like the Fibonacci numbers and it s not hard to see why The reasoning we used to establish T4 T3 T2 easily generalizes to Tn Tn Ttl2 for n 2. Thus we have the same recurrence here as for the Fibonacci numbers except that the initial values To 1 and T 1 are a little different. But these initial values are the consecutive Fibonacci numbers Fl and F2 so the T s are just Fibonacci numbers shifted up one place Tn Fn 1 for n 0. We consider this to be a closed form for Tn because the Fibonacci numbers are important enough to be considered known Also Fn itself has a closed form in terms of algebraic operations. Notice that this equation confirms the wisdom of setting To 1. But what does all this have to do with generating functions Well we re about to get to that -there s another way to figure out what Tn is. This new lb boldly go way is based on a bold idea. Let s consider the sum of all possible 2 X n where no tiling has tilings for all n 0 and call it T gone before. T l D m B HL tB HH- . The first term T on the right stands for the null .
