The combination proof of Fibonacci sequence character

Series 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Called the Fibonacci sequence. This sequence has a lot of magical properties, one of which is that the square of each Fibonacci number is exactly 1 compared to the product of the two Fibonacci number before and after it. Specifically, if the nth Fibonacci count is the Fn, then there are:

Fn+1 FN+1-FN fn+2 = ( -1) n

The Fibonacci number has a lot of combinatorial mathematical meanings. For example, using 1x1 and 1x2 bricks to cover a 1xn chessboard, the total number of programs is exactly fn+1. Shows some examples of the smaller n:

The reason behind this rule is simple: given a chessboard of length n, the coverage scheme can be divided into two categories, the last side is a 1x1 block, or the last side is a 1x2 block. The number of scenarios in the former case is entirely dependent on the number of coverage schemes for the former n-1 lattice, and the scheme number in the latter case equals the number of coverage schemes for the former n-2 lattice. Therefore, if f (n) is used to represent the number of coverage schemes for a 1xn checkerboard, then there is exactly f (n) = f (n-1) + f (n-2). In addition, since f (1) = 1, f (2) = 2, thus the next number F (3), F (4), F (5), ... It just happens to form the Fibonacci series.

In this case, the two independent 1xn chessboard is covered with bricks, and the total scheme number is fn+1 Fn+1. However, we intentionally put the two independent boards as shown on the left. Similarly, the total scheme number of a 1x (n+1) Checkerboard with a block and another 1x (n-1) chessboard is FN Fn+2, let's put the two boards as shown on the right. There is a very clever one by one correspondence between the overlay scheme of the left figure and the overlay scheme on the right.

For any of the overlay schemes in the left-hand image, we find the split line coincident across the top and bottom of the board, selecting the right-most common split line, and swapping the portion to the right of the split line. In this way, each overlay of the checkerboard on the left will turn into an overlay scheme for the right-hand chart board. According to the same method, each overlay scheme on the right-hand chart can also be changed back to the overlay scheme of the left chart board, which illustrates the fn+1 Fn+1 and FN Fn+2 is quite the same.

Wait, then why, when N is even, fn+1 Fn+1 than FN Fn+2 is 1 larger, and n is odd, fn+1 Fn+1 will be better than FN. Fn+2 Little 1? God is here. This is because the one by one correspondence that you just mentioned is not really exactly one by one corresponding. When n is an even number, there is an extremely special coverage scheme on the left-hand chart that cannot correspond to the checkerboard on the right-because there is no coincident split line in this scenario. When n is odd, the checkerboard on the left chart no longer has such a special coverage scheme, but in the right-hand chart There is a case where there is no overlap of the split line. That proves the fn+1. FN+1-FN fn+2 = ( -1) n.

The combination proof of Fibonacci sequence character