遥远的相似性——异或运算

前两天做校内题的时候突然发现模意义下的组合数和一个奇妙序列完全相同。

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模 \(2\) 意义下的组合数表：

1
11
101
1111
10001
110011
1010101
11111111
100000001
1100000011
10100000101
111100001111
1000100010001
11001100110011
101010101010101
1111111111111111
10000000000000001
110000000000000011
1010000000000000101
11110000000000001111
100010000000000010001
1100110000000000110011
10101010000000001010101
111111110000000011111111
1000000010000000100000001
11000000110000001100000011
101000001010000010100000101
1111000011110000111100001111
10001000100010001000100010001
110011001100110011001100110011
1010101010101010101010101010101
11111111111111111111111111111111
100000000000000000000000000000001
1100000000000000000000000000000011
10100000000000000000000000000000101
111100000000000000000000000000001111
1000100000000000000000000000000010001
11001100000000000000000000000000110011
101010100000000000000000000000001010101
1111111100000000000000000000000011111111
10000000100000000000000000000000100000001
110000001100000000000000000000001100000011
1010000010100000000000000000000010100000101
11110000111100000000000000000000111100001111
100010001000100000000000000000001000100010001
1100110011001100000000000000000011001100110011
10101010101010100000000000000000101010101010101
111111111111111100000000000000001111111111111111
1000000000000000100000000000000010000000000000001
11000000000000001100000000000000110000000000000011
101000000000000010100000000000001010000000000000101
1111000000000000111100000000000011110000000000001111
10001000000000001000100000000000100010000000000010001
110011000000000011001100000000001100110000000000110011
1010101000000000101010100000000010101010000000001010101
11111111000000001111111100000000111111110000000011111111
100000001000000010000000100000001000000010000000100000001
1100000011000000110000001100000011000000110000001100000011
10100000101000001010000010100000101000001010000010100000101
111100001111000011110000111100001111000011110000111100001111
1000100010001000100010001000100010001000100010001000100010001
11001100110011001100110011001100110011001100110011001100110011
101010101010101010101010101010101010101010101010101010101010101
1111111111111111111111111111111111111111111111111111111111111111

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一个全零的序列，只有第一位为 1，每一次保持第一位不变，其他位全都异或上前一位，会得到如下的序列：

1
11
101
1111
10001
110011
1010101
11111111
100000001
1100000011
10100000101
111100001111
1000100010001
11001100110011
101010101010101
1111111111111111
10000000000000001
110000000000000011
1010000000000000101
11110000000000001111
100010000000000010001
1100110000000000110011
10101010000000001010101
111111110000000011111111
1000000010000000100000001
11000000110000001100000011
101000001010000010100000101
1111000011110000111100001111
10001000100010001000100010001
110011001100110011001100110011
1010101010101010101010101010101
11111111111111111111111111111111
100000000000000000000000000000001
1100000000000000000000000000000011
10100000000000000000000000000000101
111100000000000000000000000000001111
1000100000000000000000000000000010001
11001100000000000000000000000000110011
101010100000000000000000000000001010101
1111111100000000000000000000000011111111
10000000100000000000000000000000100000001
110000001100000000000000000000001100000011
1010000010100000000000000000000010100000101
11110000111100000000000000000000111100001111
100010001000100000000000000000001000100010001
1100110011001100000000000000000011001100110011
10101010101010100000000000000000101010101010101
111111111111111100000000000000001111111111111111
1000000000000000100000000000000010000000000000001
11000000000000001100000000000000110000000000000011
101000000000000010100000000000001010000000000000101
1111000000000000111100000000000011110000000000001111
10001000000000001000100000000000100010000000000010001
110011000000000011001100000000001100110000000000110011
1010101000000000101010100000000010101010000000001010101
11111111000000001111111100000000111111110000000011111111
100000001000000010000000100000001000000010000000100000001
1100000011000000110000001100000011000000110000001100000011
10100000101000001010000010100000101000001010000010100000101
111100001111000011110000111100001111000011110000111100001111
1000100010001000100010001000100010001000100010001000100010001
11001100110011001100110011001100110011001100110011001100110011
101010101010101010101010101010101010101010101010101010101010101
1111111111111111111111111111111111111111111111111111111111111111

完 全 一 致

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事实上，二者的原理十分简单。考虑到组合数递推式：

\[\binom nk=\binom {n-1}{k-1}+\binom {n-1}{k} \]

在模 \(2\) 意义下，加法就直接成为了异或，这就使得两个序列惊人的相同。