[从头学数学] 第97节 数学广角--数与形

剧情提要：
[机器小伟]在[工程师阿伟]的陪同下进入练气期第十一层功法的修炼，
这次要修炼的目标是[数学广角--数与形]。

正剧开始：

星历2016年02月19日 16:32:09, 银河系厄尔斯星球中华帝国江南行省。
[工程师阿伟]正在和[机器小伟]一起研究数与形。

其实这就是找规律啊，小伟对于找规律是最有兴趣的了。

<span style="font-size:18px;">>>> 
1+3+5+7+9 = 25 = 5 * 5
1+3+5+7+9+11 = 36 = 6 * 6
1+3+5+7+9+11+13 = 49 = 7 * 7
1+3+5+7+9+11+13+15 = 64 = 8 * 8
1+3+5+7+9+11+13+15+17 = 81 = 9 * 9

def tmp():
    begin = 1;
    end = 10;
    s = '';
    total = 0;

    for j in range(10, 20, 2):
        begin = 1;
        end = j;
        s = '';
        total = 0;
        for i in range(begin, end, 2):
            s += str(i)+'+';
            total += i;

        sqr = int(math.sqrt(total));
        print('{0} = {1} = {2} * {2}'.format(s[:-1], total, sqr, sqr));
    return;</span>

<span style="font-size:18px;">>>> 
1/4+1/8  =  3/4
1/4+1/8+1/16  =  7/8
1/4+1/8+1/16+1/32  =  15/16
1/4+1/8+1/16+1/32+1/64  =  31/32
1/4+1/8+1/16+1/32+1/64+1/128  =  63/64
1/4+1/8+1/16+1/32+1/64+1/128+1/256  =  127/128
1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512  =  255/256
1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024  =  511/512
1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024+1/2048  =  1023/1024

def tmp():
    a = [];
    b = 2;
    c = Fraction(1, b);
    s = '';

    for j in range (1, 10):
        a = [];
        b = 2;
        c = Fraction(1, b);
        s = '';
        for i in range(j+1):
            a.append(c);
            b*=2;
            c = Fraction(1, b);
            s += str(c)+'+';
        print(s[:-1], ' = ', sum(a)); 
    return;
	</span>

这应该就是傅立叶级数以及极限的最初的露面了，其实[人叫板老师]是比较阴险的， 不算极限的话，上面这个算式不可能是1。

<span style="font-size:18px;">>>> 
1 = 1 = 2(1*1)-1
1+3+1 = 5 = 2(2*2)-3
1+3+5+3+1 = 13 = 2(3*3)-5
1+3+5+7+5+3+1 = 25 = 2(4*4)-7
1+3+5+7+9+7+5+3+1 = 41 = 2(5*5)-9
1+3+5+7+9+11+9+7+5+3+1 = 61 = 2(6*6)-11
1+3+5+7+9+11+13+11+9+7+5+3+1 = 85 = 2(7*7)-13
1+3+5+7+9+11+13+15+13+11+9+7+5+3+1 = 113 = 2(8*8)-15
1+3+5+7+9+11+13+15+17+15+13+11+9+7+5+3+1 = 145 = 2(9*9)-17

def tmp():
    begin = 1;
    end = 10;
    s = '';
    total = 0;

    for j in range(2, 20, 2):
        begin = 1;
        end = j;
        s = '';
        total = 0;
        for i in range(begin, end, 2):
            s += str(i)+'+';
            total += i;

        for i in range(end-3, begin-1, -2):
            s += str(i)+'+';
            total += i;
        
        print('{0} = {1} = 2({2}*{2})-{3}'.format(s[:-1], total, end//2, end-1));
    return;</span>

<span style="font-size:18px;">>>> 
1  -->  red: 1 blue: 8
2  -->  red: 2 blue: 10
3  -->  red: 3 blue: 12
4  -->  red: 4 blue: 14
5  -->  red: 5 blue: 16
6  -->  red: 6 blue: 18
7  -->  red: 7 blue: 20
8  -->  red: 8 blue: 22
9  -->  red: 9 blue: 24
10  -->  red: 10 blue: 26
11  -->  red: 11 blue: 28
12  -->  red: 12 blue: 30
13  -->  red: 13 blue: 32
14  -->  red: 14 blue: 34

def tmp():
    for n in range(1, 15):
        red = n;
        blue = (n+2)*3-n;
        print(n, ' --> ', 'red:', red, 'blue:', blue);
    return;
        </span>

<span style="font-size:18px;">>>> 
1  -->  8
2  -->  16
3  -->  24
4  -->  32
5  -->  40
6  -->  48
7  -->  56
8  -->  64
9  -->  72
10  -->  80
11  -->  88
12  -->  96
13  -->  104
14  -->  112
>>> 

#题1
def tmp():
    for n in range(1, 15):
        a = (2*n+1)*(2*n+1)- (2*n-1)*(2*n-1);
        print(n, ' --> ',a);
    return;</span>

<span style="font-size:18px;">>>> 
1  -->  1
2  -->  3
3  -->  6
4  -->  10
5  -->  15
6  -->  21
7  -->  28
8  -->  36
9  -->  45
10  -->  55
11  -->  66
12  -->  78
13  -->  91
14  -->  105

#题2
def tmp():
    a = []
    for n in range(1, 15):
        a.append(n);
        print(n, ' --> ',sum(a));
    return;</span>

<span style="font-size:18px;">>>> 
1  -->  n: 1 C: 3
4  -->  n: 4 C: 6
9  -->  n: 9 C: 9
16  -->  n: 16 C: 12
25  -->  n: 25 C: 15
36  -->  n: 36 C: 18
49  -->  n: 49 C: 21
64  -->  n: 64 C: 24
81  -->  n: 81 C: 27
100  -->  n: 100 C: 30
121  -->  n: 121 C: 33
144  -->  n: 144 C: 36
169  -->  n: 169 C: 39
196  -->  n: 196 C: 42

#题3
def tmp():
    n = 0;
    for i in range(1, 15):
        n += (2*i-1);
        C = 3*i;
        print(n, ' --> ','n:', n, 'C:', C);
    return;</span>

<span style="font-size:18px;">>>> 
             [1]              
            [1, 1]            
          [1, 2, 1]           
         [1, 3, 3, 1]         
       [1, 4, 6, 4, 1]        
     [1, 5, 10, 10, 5, 1]     
   [1, 6, 15, 20, 15, 6, 1]   
 [1, 7, 21, 35, 35, 21, 7, 1] 
[1, 8, 28, 56, 70, 56, 28, 8, 1]
[1, 9, 36, 84, 126, 126, 84, 36, 9, 1]

#题7
def tmp():
    res = [[1]];
    n = 10;
    width = n*3;
    
    for i in range(1, n):  
        a = [];  
        for j in range(i+1):  
            if (j == 0 or j == i):  
                a.append(res[i-1][0]);  
            else:  
                a.append(res[i-1][j-1]+res[i-1][j]);  
        res.append(a);

    for i in range(n):
        print('{0:^{w}}'.format(res[i], w=width));
    return;</span>

本节到此结束，欲知后事如何，请看下回分解。