10884 - Persephone

困扰了很久， 偶尔看到了公式， 记下。

Expanding the generating function shows that the number of convex polyominoes having perimeter is given by

where , , and is a binomial coefficient。

公式来源于这里： http://mathworld.wolfram.com/ConvexPolyomino.html

import java.math.BigInteger;
import java.util.Scanner;


public class Main {

    static BigInteger mypow(int n, int k) {
        BigInteger ans = BigInteger.ONE;
        BigInteger N = BigInteger.valueOf(n);
        for(; k != 0; k >>= 1) {
            if((k & 1) != 0) ans = ans.multiply(N);
            N = N.multiply(N);
        }
        return ans;
    }
    static BigInteger C(int n, int k) {
        BigInteger ans = BigInteger.ONE;
        for(int i = 0; i < k; ++i) {
            ans = ans.multiply(BigInteger.valueOf(n - i)).divide(BigInteger.valueOf(i + 1));
        }
        return ans;
    }
    static BigInteger solve(int n) {
        BigInteger A = BigInteger.valueOf(2 * n + 11).multiply(mypow(4, n));
        //System.out.println("A = " + A);
        BigInteger B = BigInteger.valueOf(4 * (2 * n + 1)).multiply(C(2 * n, n));
        return A.subtract(B);
    }
    public static void main(String[] args) {
        Scanner in = new Scanner(System.in);
        int T = in.nextInt();
        for(int kase = 1; kase <= T; ++kase) {
            int n = in.nextInt();
            System.out.print("Case #" + kase + ": ");
            if(((n & 1) != 0) || (n == 2)) {
                System.out.println("0");
            } else if(n == 4) {
                System.out.println("1");
            } else if(n == 6) {
                System.out.println("2");
            } else if(n == 8) {
                System.out.println("7");
            } else {
                n = (n - 8) / 2;
                System.out.println(solve(n));
            }
        }

    }

}