【算法笔记】数学·杂

利用Pentagonal Numbers（五边形数定理）计算整数拆分

you should know:
1）分割函数P(n)，简单地说就是n的无序拆分数
2）生成函数

相关资料
see http://blog.youkuaiyun.com/acdreamers/article/details/12259815
also http://pages.uoregon.edu/koch/PentagonalNumbers.pdf
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如果不想知道原理，可以这样理解，五边形数定理给出了通过P(n)前面某些项（不大于n）的和差来求P(n)的方法。
广义五边形数的通项公式 $(-1)^kk(3k^+_-1)/2$，k从1开始
-1, -2, 5, 7, -12….
两个负号，两个正号交替出现
P(13) 就可以表示为 P(13 - 12), P(13 - 7), P(13 - 5), P(13 - 2), P(13 - 1)的和差
P(13) = P(1) - P(5) - P(7) + P(11) + P(12)
发现了吗，加上还是减去P(n-k)和k在数列中的符号是相反的

因为计算第n项，要用到所有比n小的5边形数，大概有$\sqrt{n}$个所以时间复杂度是 O(nsqrt(n))

例题 51nod 1259

import java.io.InputStream;
import java.io.InputStreamReader;
import java.io.BufferedReader;
import java.io.OutputStream;
import java.io.PrintWriter;
import java.io.IOException;
import java.io.File;
import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.util.StringTokenizer;
import java.math.*;

public class Main {

    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;

        try {
            inputStream = new FileInputStream(new File("in.txt"));
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }

        InputReader in = new InputReader(inputStream);
        PrintWriter out = new PrintWriter(outputStream);

        Solver sol = new Solver();

        sol.solve(in, out);

        out.close();
    }
}

// Pentagonal number theorem
// see http://blog.youkuaiyun.com/acdreamers/article/details/12259815
// also http://pages.uoregon.edu/koch/PentagonalNumbers.pdf
class Solver {
    final static int Mod = (int)(1e9 + 7);
    final static int Maxn = 50000;

    static int tab[]  = new int [400+5];  // tab of pentagonal number
    static int p[]    = new int [Maxn+5]; // tab of P(k)

    public void solve(InputReader in, PrintWriter out) {
        // init tab
        int tot = 1;
        for (int i=1;tot <= 400;++i) {
            tab[tot ++] = i * (3*i-1) / 2;
            tab[tot ++] = i * (3*i+1) / 2;
        }

        p[0] = 1; p[1] = 1;
        int n = in.nextInt();
        for (int i=2;i<=n;++i) {
            int ans = 0;
            for (int j=1;tab[j] <= i;++j) {
                if ( ((j+1) >> 1 & 1) > 0)
                    ans = ans + p[i - tab[j]];
                else
                    ans = ans - p[i - tab[j]];
                if (ans < 0) ans += Mod;
                else if (ans >= Mod) ans -= Mod;
            }
            p[i] = ans;
        }
        out.println(p[n]);
    }
}

class InputReader {
    public BufferedReader reader;
    public StringTokenizer tokenizer;

    public InputReader(InputStream stream) {
        reader = new BufferedReader(new InputStreamReader(stream), 32768);
        tokenizer = null;
    }

    public String next() {
        while (tokenizer == null || !tokenizer.hasMoreTokens()) {
            try {
                tokenizer = new StringTokenizer(reader.readLine());
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }
        return tokenizer.nextToken();
    }

    public boolean hasNext() {
    while (tokenizer == null || !tokenizer.hasMoreTokens()) {
            try {
            String line = reader.readLine();
        if (line == null) return false;
                tokenizer = new StringTokenizer(line);
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }
    return true;
    }

    public int nextInt() {
        return Integer.parseInt(next());
    }

}