演算法筆記_093:藍橋杯練習 Problem S4: Interesting Numbers 加強版（Java）

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目錄

1 問題描述

2 解決方案

 1 問題描述

Problem Description
　　We call a number interesting, if and only if:
　　1. Its digits consists of only 0, 1, 2 and 3, and all these digits occurred at least once.
　　2. Inside this number, all 0s occur before any 1s, and all 2s occur before any 3s.
　　Therefore, the smallest interesting number according to our definition is 2013. There are two more interseting number of 4 digits: 2031 and 2301.
　　Your task is to calculate the number of interesting numbers of exactly n digits. As the answer might be very large, you only need to output the answer modulo 1000000007.
Input Format
　　The input has one line consisting of one positive integer n (4 ≤ n ≤ 10^15).
Output Format
　　The output has just one line, containing the number of interesting numbers of exactly n digits, modulo 1000000007.
Input Sample
　　4
Output Sample
　　3

 2 解決方案

本題主要考查數學組合數推理化簡，具體思考過程如下：

引用自文末參考資料1：

 

推導過程，在草稿紙上推導了一下：

本文下面代碼結果運行為90分，代碼僅供參考，文末參考資料1中代碼運行結果為100分，可以參考一下哦。

具體代碼如下：

import java.util.Scanner;public class Main {    public final static long p = 1000000007L;    //求取a的b次方取餘p的值    public long getPowMod(long a, long b) {        long temp = a, result = 1;        while(b != 0) {            if((b & 1) == 1)                result = result * temp % p;            temp = temp * temp % p;            b >>= 1;        }        return result;    }        public void printResult(long n) {        if(n == 4) {            System.out.println(3);            return;        }        long m = n - 1;        long result = getPowMod(2, m - 2);        m = m % p;        result = result * (m * m % p - 3 * m % p) % p + m;        result %= p;        System.out.println(result);        return;    }        public static void main(String[] args) {        Main test = new Main();        Scanner in = new Scanner(System.in);        long n = in.nextLong();        test.printResult(n);            }}

 

 

 

參考資料：

1. 演算法提高 Problem S4: Interesting Numbers 加強版 解題報告

 

演算法筆記_093:藍橋杯練習 Problem S4: Interesting Numbers 加強版（Java）