sequence puzzle

[Son of Waj]

I need this to write a math paper. Here's a sequence creator.
http://www.maths.ex.ac.uk/pumc/sequences.html

Here's what I've figured out from this one: The sum of the terms of the sequence I input is equal to the sum of the terms of the outputted sequence. Also the number of terms of my sequence is equal to the first term of the outputted sequence. I couldn't figure out much more from there.

P.S. Do you know any useful applications of knowing that a sequence is monotonic. Furthermore, do you know any applications for knowing the length of the longest monotonic subsequence within a sequence?

[This message has been edited by Son of Waj (edited 04-14-2002 03:47 PM).]

[CrystyB]

Yours 10 8 8 6 3 3 2 1
Mine 8 7 6 4 4 4 3 3 1 1

Yours 10 9 8 6 3 3 2 1
Mine 8 7 6 4 4 4 3 3 2 1

Yours 10 8 8 6 4 3 2 1
Mine 8 7 6 5 4 4 3 3 1 1

Yours 10 9 8 6 4 3 2 1
Mine 8 7 6 5 4 4 3 3 2 1

Yours ... 5 4 3 2 1
Mine ... 5 4 3 2 1

Yours 5 5 5 1
Mine 4 3 3 3 3

Yours 5 5 5 3 3 1
Mine 6 5 5 3 3

Yours 15 5 5 3 3 1
Mine 6 5 5 3 3 1 1 1 1 1 1 1 1 1 1

note that it works both ways. Also, take the first 5 there are only 1's in the answer; after the first 3 there are some 3's before the 1's. Let me elaborate:

Yours 9 6 6 6 4 4 2 2
Mine 8 8 6 6 4 4 1 1 1

Eliminating the repetitions of the answer, it counts the terms of prefixes that end with the last appearance of the digit: 9 has 1 term, 9 & 6's has 4 terms, 9 & 6's & 4's has 6 and the whole series has 8.
As for the number of repetitions, it is the DIFFERENCE between the previous last term and the current one (9-6=3, 6-4=2, 4-2=2), with the first term repeated L times (L=last term in given seq). Here:

Yours 12 12 12 8 8 5
Mine 6 6 6 6 6 5 5 5 3 3 3 3

Now someone please make these rules of mine a little more beautiful? TYIA!

[mathgrant]

It was easy. 6 4 3 1:

code:

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Go down the columns and count: 4 3 3 2 1 1