【Java】斐波那契数列（Fibonacci Sequence、兔子数列）的3种计算方法（递归实现、递归值缓存实现、循环实现、尾递归实现）

斐波那契数列：0、1、1、2、3、5、8、13…………

他的规律是，第一项是0，第二项是1，第三项开始（含第三项）等于前两项之和。

> 递归实现

看到这个规则，第一个想起当然是递归算法去实现了，于是写了以下一段：

public class RecursionForFibonacciSequence {

    public static void main(String[] args) {
        System.out.println(recursion(10));
    }
    
    public static double recursion(double i) {
        if (i == 0) {
            printResult(i, 0);
            return 0;
        }
        if (i == 1) {
            printResult(i, 1);
            return 1;
        }
        
        double result = recursion(i - 1) + recursion(i - 2);
        printResult(i, result);
        
        return result;
    }
    
    /**
     * 打印结果
     */
    public static void printResult(double i, double result) {
        System.out.println(i + " -> " + result);
    }
    
}

它能正常运行，比如计算第10项的结果为55。

但是，计算数字大点的数据，则很慢很慢，因为重复计算太多了。

日志：

1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
6.0 -> 8.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
7.0 -> 13.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
6.0 -> 8.0
8.0 -> 21.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
6.0 -> 8.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
7.0 -> 13.0
9.0 -> 34.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
6.0 -> 8.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
7.0 -> 13.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
5.0 -> 5.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
4.0 -> 3.0
6.0 -> 8.0
8.0 -> 21.0
10.0 -> 55.0
55.0

> 递归值缓存实现

用最直观的方式优化，既然重复计算太多了，而重复计算的结果都是一样的，那么我们就将重复计算的结果集缓存起来吧。

因为上例的递归效率低，不能执行太多的项数，所以只执行到10，而下面这个写法的效率大为提高，所以我们执行到100看看。

import java.util.HashMap;
import java.util.Map;

public class CacheForFibonacciSequence {

    public static void main(String[] args) {
        System.out.println(recursion(100));
    }
    
    // 缓存计算结果集
    public static Map<Double, Double> map = new HashMap<Double, Double>();
    
    public static double recursion(double i) {
        if (i == 0) {
            printResult(i, 0);
            return 0;
        }
        if (i == 1) {
            printResult(i, 1);
            return 1;
        }
        
        if (map.containsKey(i)) {
            return map.get(i);
        }
        
        double result = recursion(i - 1) + recursion(i - 2);
        printResult(i, result);
        map.put(i, result);
        
        return result;
    }
    
    /**
     * 打印结果
     */
    public static void printResult(double i, double result) {
        System.out.println(i + " -> " + result);
    }
    
}

日志：

1.0 -> 1.0
0.0 -> 0.0
2.0 -> 1.0
1.0 -> 1.0
3.0 -> 2.0
4.0 -> 3.0
5.0 -> 5.0
6.0 -> 8.0
7.0 -> 13.0
8.0 -> 21.0
9.0 -> 34.0
10.0 -> 55.0
11.0 -> 89.0
12.0 -> 144.0
13.0 -> 233.0
14.0 -> 377.0
15.0 -> 610.0
16.0 -> 987.0
17.0 -> 1597.0
18.0 -> 2584.0
19.0 -> 4181.0
20.0 -> 6765.0
21.0 -> 10946.0
22.0 -> 17711.0
23.0 -> 28657.0
24.0 -> 46368.0
25.0 -> 75025.0
26.0 -> 121393.0
27.0 -> 196418.0
28.0 -> 317811.0
29.0 -> 514229.0
30.0 -> 832040.0
31.0 -> 1346269.0
32.0 -> 2178309.0
33.0 -> 3524578.0
34.0 -> 5702887.0
35.0 -> 9227465.0
36.0 -> 1.4930352E7
37.0 -> 2.4157817E7
38.0 -> 3.9088169E7
39.0 -> 6.3245986E7
40.0 -> 1.02334155E8
41.0 -> 1.65580141E8
42.0 -> 2.67914296E8
43.0 -> 4.33494437E8
44.0 -> 7.01408733E8
45.0 -> 1.13490317E9
46.0 -> 1.836311903E9
47.0 -> 2.971215073E9
48.0 -> 4.807526976E9
49.0 -> 7.778742049E9
50.0 -> 1.2586269025E10
51.0 -> 2.0365011074E10
52.0 -> 3.2951280099E10
53.0 -> 5.3316291173E10
54.0 -> 8.6267571272E10
55.0 -> 1.39583862445E11
56.0 -> 2.25851433717E11
57.0 -> 3.65435296162E11
58.0 -> 5.91286729879E11
59.0 -> 9.56722026041E11
60.0 -> 1.54800875592E12
61.0 -> 2.504730781961E12
62.0 -> 4.052739537881E12
63.0 -> 6.557470319842E12
64.0 -> 1.0610209857723E13
65.0 -> 1.7167680177565E13
66.0 -> 2.7777890035288E13
67.0 -> 4.4945570212853E13
68.0 -> 7.2723460248141E13
69.0 -> 1.17669030460994E14
70.0 -> 1.90392490709135E14
71.0 -> 3.08061521170129E14
72.0 -> 4.98454011879264E14
73.0 -> 8.06515533049393E14
74.0 -> 1.304969544928657E15
75.0 -> 2.11148507797805E15
76.0 -> 3.416454622906707E15
77.0 -> 5.527939700884757E15
78.0 -> 8.944394323791464E15
79.0 -> 1.447233402467622E16
80.0 -> 2.3416728348467684E16
81.0 -> 3.7889062373143904E16
82.0 -> 6.1305790721611584E16
83.0 -> 9.9194853094755488E16
84.0 -> 1.60500643816367072E17
85.0 -> 2.5969549691112256E17
86.0 -> 4.2019614072748966E17
87.0 -> 6.7989163763861222E17
88.0 -> 1.10008777836610189E18
89.0 -> 1.77997941600471398E18
90.0 -> 2.880067194370816E18
91.0 -> 4.6600466103755305E18
92.0 -> 7.5401138047463465E18
93.0 -> 1.2200160415121877E19
94.0 -> 1.9740274219868226E19
95.0 -> 3.19404346349901E19
96.0 -> 5.168070885485833E19
97.0 -> 8.362114348984843E19
98.0 -> 1.3530185234470676E20
99.0 -> 2.189229958345552E20
100.0 -> 3.54224848179262E20
3.54224848179262E20

> 循环方式

还有我们也可以用循环的方式，只用两个变量缓存前两项的值：

public class ForeachForFibonacciSequence {

    public static void main(String[] args) {
        System.out.println(foreach(100));
    }
    
    public static double foreach(double i) {
        if (i <= 0d) {
            return 0d;
        }
        
        if (i == 1d) {
            return 1d;
        }
        
        double temp1 = 0d;
        double temp2 = 1d;
        double tempSum = 0;
        for (double d = 2; d <= i; d++) {
            tempSum = temp1 + temp2;
            printResult(d, tempSum);
            
            temp1 = temp2;
            temp2 = tempSum;
        }
        
        return tempSum;
    }
    
    /**
     * 打印结果
     */
    public static void printResult(double i, double result) {
        System.out.println(i + " -> " + result);
    }
    
}

日志：

2.0 -> 1.0
3.0 -> 2.0
4.0 -> 3.0
5.0 -> 5.0
6.0 -> 8.0
7.0 -> 13.0
8.0 -> 21.0
9.0 -> 34.0
10.0 -> 55.0
11.0 -> 89.0
12.0 -> 144.0
13.0 -> 233.0
14.0 -> 377.0
15.0 -> 610.0
16.0 -> 987.0
17.0 -> 1597.0
18.0 -> 2584.0
19.0 -> 4181.0
20.0 -> 6765.0
21.0 -> 10946.0
22.0 -> 17711.0
23.0 -> 28657.0
24.0 -> 46368.0
25.0 -> 75025.0
26.0 -> 121393.0
27.0 -> 196418.0
28.0 -> 317811.0
29.0 -> 514229.0
30.0 -> 832040.0
31.0 -> 1346269.0
32.0 -> 2178309.0
33.0 -> 3524578.0
34.0 -> 5702887.0
35.0 -> 9227465.0
36.0 -> 1.4930352E7
37.0 -> 2.4157817E7
38.0 -> 3.9088169E7
39.0 -> 6.3245986E7
40.0 -> 1.02334155E8
41.0 -> 1.65580141E8
42.0 -> 2.67914296E8
43.0 -> 4.33494437E8
44.0 -> 7.01408733E8
45.0 -> 1.13490317E9
46.0 -> 1.836311903E9
47.0 -> 2.971215073E9
48.0 -> 4.807526976E9
49.0 -> 7.778742049E9
50.0 -> 1.2586269025E10
51.0 -> 2.0365011074E10
52.0 -> 3.2951280099E10
53.0 -> 5.3316291173E10
54.0 -> 8.6267571272E10
55.0 -> 1.39583862445E11
56.0 -> 2.25851433717E11
57.0 -> 3.65435296162E11
58.0 -> 5.91286729879E11
59.0 -> 9.56722026041E11
60.0 -> 1.54800875592E12
61.0 -> 2.504730781961E12
62.0 -> 4.052739537881E12
63.0 -> 6.557470319842E12
64.0 -> 1.0610209857723E13
65.0 -> 1.7167680177565E13
66.0 -> 2.7777890035288E13
67.0 -> 4.4945570212853E13
68.0 -> 7.2723460248141E13
69.0 -> 1.17669030460994E14
70.0 -> 1.90392490709135E14
71.0 -> 3.08061521170129E14
72.0 -> 4.98454011879264E14
73.0 -> 8.06515533049393E14
74.0 -> 1.304969544928657E15
75.0 -> 2.11148507797805E15
76.0 -> 3.416454622906707E15
77.0 -> 5.527939700884757E15
78.0 -> 8.944394323791464E15
79.0 -> 1.447233402467622E16
80.0 -> 2.3416728348467684E16
81.0 -> 3.7889062373143904E16
82.0 -> 6.1305790721611584E16
83.0 -> 9.9194853094755488E16
84.0 -> 1.60500643816367072E17
85.0 -> 2.5969549691112256E17
86.0 -> 4.2019614072748966E17
87.0 -> 6.7989163763861222E17
88.0 -> 1.10008777836610189E18
89.0 -> 1.77997941600471398E18
90.0 -> 2.880067194370816E18
91.0 -> 4.6600466103755305E18
92.0 -> 7.5401138047463465E18
93.0 -> 1.2200160415121877E19
94.0 -> 1.9740274219868226E19
95.0 -> 3.19404346349901E19
96.0 -> 5.168070885485833E19
97.0 -> 8.362114348984843E19
98.0 -> 1.3530185234470676E20
99.0 -> 2.189229958345552E20
100.0 -> 3.54224848179262E20
3.54224848179262E20

> 尾递归方式

从回复的网友“Mr_listening”的博文中了解到，还可以用尾递归的方式实现，看以下代码：

public class RecursionTailForFibonacciSequence {

    public static void main(String[] args) {
        System.out.println(recursionTail(10, 1, 1));
    }
    
    public static double recursionTail(int i, double temp1, double temp2) {
        if (i < 2) {
            return temp1;
        }
        
        double result = recursionTail(i - 1, temp2, temp1 + temp2);
        printResult(i - 1, temp2, temp1 + temp2, result);
        return result;
    }
    
    /**
     * 打印结果
     */
    public static void printResult(double i, double temp1, double temp2, double result) {
        System.out.println("recursionTail(" + i + ", " + temp1 + ", " + temp2 + ") -> " + result);
    }
    
}

日志：

recursionTail(1.0, 55.0, 89.0) -> 55.0
recursionTail(2.0, 34.0, 55.0) -> 55.0
recursionTail(3.0, 21.0, 34.0) -> 55.0
recursionTail(4.0, 13.0, 21.0) -> 55.0
recursionTail(5.0, 8.0, 13.0) -> 55.0
recursionTail(6.0, 5.0, 8.0) -> 55.0
recursionTail(7.0, 3.0, 5.0) -> 55.0
recursionTail(8.0, 2.0, 3.0) -> 55.0
recursionTail(9.0, 1.0, 2.0) -> 55.0
55.0