We don't bite newbies here... much PerlMonks

### comment on

 Need Help??
Let's examine the properties of the sequence:

### It is made up only of 1's, 2's, and 3's

We can prove this by contradiction. Because the sequence starts with '1', the only way a '4' can show up is as the first number in a pair (the "run" value -- the bold values in 312211). After all, that's the only way that the '2' and '3' ever show up. So this means we must have a sub-sequence such as "1111", "2222", or "3333" in our sequence. Let's abstract these as "xxxx".

There are two ways "xxxx" can be placed in the sequence, at an even offset or an odd offset. At an even offset, the first and third 'x's are counts; at an odd offset, the second and third 'x's are counts. Let's examine the the even offset first. You can't have "C1xC2x" in the sequence, because that means it should have been encoded as "(C1+C2)x". Similarly, at an odd offset, there must be a count before the first 'x' (we'll call it C1 again), which means we have "C1xxxx" in our sequence. Again, you can't have two counts in a row for the same value! The subsequence would have to be "(C1+x)xx".

So this means there will never be four like values in a row, thus '4' will never be in this sequence. (You can prove that '2' and '3' WILL be in the sequence.)

Every span of like values results in a two-character sequence (count and value). 1 value in a row results in a gain of a character, 2 values in a row results in no change, and 3 values in a row results in a loss of 1 character. What we have to examine is the sequence and show that triplets are far less common than singlets and couplets.
_____________________________________________________
Jeff japhy Pinyan, P.L., P.M., P.O.D, X.S.: Perl, regex, and perl hacker
How can we ever be the sold short or the cheated, we who for every service have long ago been overpaid? ~~ Meister Eckhart

In reply to Re^3: Conway's audioactive sequence oneliner by japhy
in thread Conway's audioactive sequence oneliner by blazar

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post; it's "PerlMonks-approved HTML":

• Are you posting in the right place? Check out Where do I post X? to know for sure.
• Posts may use any of the Perl Monks Approved HTML tags. Currently these include the following:
<code> <a> <b> <big> <blockquote> <br /> <dd> <dl> <dt> <em> <font> <h1> <h2> <h3> <h4> <h5> <h6> <hr /> <i> <li> <nbsp> <ol> <p> <small> <strike> <strong> <sub> <sup> <table> <td> <th> <tr> <tt> <u> <ul>
• Snippets of code should be wrapped in <code> tags not <pre> tags. In fact, <pre> tags should generally be avoided. If they must be used, extreme care should be taken to ensure that their contents do not have long lines (<70 chars), in order to prevent horizontal scrolling (and possible janitor intervention).
• Want more info? How to link or or How to display code and escape characters are good places to start.

Create A New User
Domain Nodelet?
Chatterbox?
and the web crawler heard nothing...

How do I use this? | Other CB clients
Other Users?
Others wandering the Monastery: (2)
As of 2021-12-05 12:04 GMT
Sections?
Information?
Find Nodes?
Leftovers?
Voting Booth?
R or B?

Results (31 votes). Check out past polls.

Notices?