Can a sequence simplified to one number? Is there any algorithm there can convert any (non-infinity) sequence into one number (and back again with one solution)?
Of course can it be turned into a N-base number (N=max(sequence)+1) but it require the highest number in the sequence which have multiple solutions.
Thank you.
Sorry for my bad english
 A: Basically, you can always string the sequence together. You just need a special character to act as a delimiter. This can be taken care of by using bases, as you suggest.
Here is an algorithm for converting a sequence into a number:


*

*Assume that the sequence $a_1, ... ,a_n$ is given in base 9.

*Write $$N = a_1.9.a_2.9. a_3 ...  9.a_n . $$
Here "." means just putting the digits one after the other. $N$ should be interpreted as a number in base 10.
To convert a number into a sequence, just take $a_1$ to be all the digits until the first $9$, take $a_2$ to be the digits between the first $9$ and the second $9$, and so on. The elements of the sequence should be interpreted as numbers in base 9, of course.
A: I suggest continued fractions.
http://en.wikipedia.org/wiki/Continued_fraction
For example, if your sequence is $1,2,3,\dots$, it would become
$$\dfrac1{1+\dfrac1{2+\dfrac1{3+\dfrac1{4+\dots}}}}=0.69777465796400798201\dots.$$
See Wolfram Alpha calculation.
And the reverse calculation to recover the sequence.
