Preface (running account): It began quite simply, in terms of biology class to the principle of complementary base pairing, spoke without limiting the number of base pairs in the case, \ (the n-\) base pairs of DNA sequence has a theoretical composition \ (4 ^ n \) species. At first glance, \ (the n-4 ^ \) from one end of the beginning of the enumeration a single chain, the multiplication principle to explain what seems to be no problem, but I am careful students ( BeyondLimits key figures ) find things not quite right, that the number should again divided by \ (2 \) , namely \ (\ the n-FRAC {4} ^ {2} \) , but I felt like I was not quite right, after discussion, we feel that this suspect is unlikely to know later found ... there is also the problem of the almost discussion ( link ), but did not give a clear conclusion.
We began to think that a DNA sequence can be calculated \ (2 \) times. However, the push \ (2 \) a number of base pairs of DNA sequence, detecting and \ (\ frac {4 ^ n } {2} \) conjecture does not comply. We also take into account the special nature of the strings, we began to consider some strings with special properties, and wrote some of the wrong program to simulate the process of enumeration procedures ... after all, is wrong, toss a long time have no progress.
From noon to make mistakes all the way into the night, early morning and we have another student ( Thorn key figure ) to discuss. And some Scrapped, just come up with the results of this question.
Play table found:
\ (n-2 = \) the DNA sequences when (10 \) \ species
\ (n-=. 3 \) the DNA sequences when (32 \) \ species
\ (n-=. 4 \) the DNA sequences when (136 \) \ species
\ (n-=. 5 \) the DNA sequences when (512 \) \ species
\ (n-=. 6 \) the DNA sequences when (2080 \) \ species
\(...\)
Found that only \ (n \) is odd to meet this conjecture, the even is not.
Back to the original point of view, we believe that a DNA sequence can be calculated \ (2 \) times. because:
But the results do not match to guess, we guess there may be other times of string is not calculated to \ (2 \) times.
There is to be greater than count the number \ (2 \) string it? It does not exist, each DNA from the 3 'end of both strands started only two results.
However, both the result may be the same?
If we follow the \ (4 ^ n \) from the 3 'end of the nucleotide one by one in the order enumerated on a single stranded DNA, there will be such a string \ (S \) from both 3' end of the start the same results, only to be enumerated \ (1 \) times.
As shown is a string \ (S \) Examples:
且因为这个特殊性质，串 \(S\) 的长度只能为偶数，因为若长度为奇数，则最中间的碱基从两个 3‘ 端开始枚举必然是不相同的。
还有一个问题就是怎么计算 DNA 序列的数目。
在枚举到的 \(4^n\) 种情况中，有些事被枚举到两次的串，有些则是只被枚举到一次的串。
只要计算出只被枚举到一次的串的数量就可以算出偶数个碱基对组成的 DNA 序列理论上的种类了。
可以按照如图方式构造：
对于 \(n\)（偶数）个碱基对 DNA 序列中，有 \(4^{\frac{n}{2}}\) 种串（序列）只会被统计到一次。
\(n\)（奇数）个则每个串（序列）都被统计两次。
所以结论是这样
\(a_n=\begin{cases}\frac{4^n}{2}\ \ \ \ \ \ \ \ n=2k+1\\\frac{4^n+4^{\frac{n}{2}}}{2}\ n=2k\ \ (k\in N^*)\end{cases}\)
但是事实上 DNA 更复杂，这个结论也只能是理论上的了。
最后感谢两位朋友凌晨一起讨论问题
@BeyondLimits
@thorn