![]()
1026: The Staircases
| Result | TIME Limit | MEMORY Limit | Run Times | AC Times | JUDGE |
|---|---|---|---|---|---|
 |
3s | 8192K | 1590 | 530 | Standard |
One curious child has a set of N little bricks. From these bricks he builds different staircases. Staircase consists of steps of different sizes in a strictly descending order. It is not allowed for staircase to have steps equal sizes. Every staircase consists of at least two steps and each step contains at least one brick. Picture gives examples of staircase for N=11 and N=5:
Your task is to write a program that reads from input numbers N and writes to output numbers Q - amount of different staircases that can be built from exactly N bricks.
Input
Numbers N, one on each line. You can assume N is between 3 and 500, both inclusive. A number 0 indicates the end of input.
Output
Numbers Q, one on each line.
Sample Input
3 5 0
Sample Output
1 2
這道題可以用動態規劃和組合數學的思想，這裏只說組合數學的思想，主要就是母函數的思想，輸入N，那麼此時有N個磚塊來組成樓梯，我個人理解其實就是把樓梯可以分解成n個有不同磚塊數目組成的長方形和正方形，如圖中N=5時，有兩種分解方法，第一種是分解爲一個磚頭組成的正方形，和四個方塊組成的長方形，第二種是將其分解爲一個磚頭組成的正方形和四個磚塊組成的長方形。並且每輸入一個N，那麼一定可以按照這種方法分解，那麼這個道題就可以變成一道組合計數的問題，從而可以想到用母函數的思想來解決，(1+x^1)(1+x^2).....(1+x^N),算出x^N項的係數，然後再減1，即爲所求的解，例如，當N=5的時候，(1+x^1)(1+x^2)(1+x^3)(1+x^4)(1+x^5)，求出x^5的係數爲3，x^5=1(x^0)*x^5=x^1*x4=x^2*x^3，而第一種1*x^5是不符合情況的，因爲，這種相當於只有一個由5個磚塊組成的長方形，想象一下，這就是1個樓梯，不符合題意，所以要將這個情況減去。
代碼：
語言：c++