Originally posted by humy
On reflection, I think the table would be slightly less likely to lead to misunderstand if presented thus:
_____| v=0 _| v=1 _| v=2 _| v=3 _| v=4 _| v=5 _| …
ov=0 ___ 3 ___ 1 ___ 1 ___ 1 ____1 ____1 __ ...
ov=1 ___ 3 ___ 4 ___ 5 ___ 6 ____7 ____8 __...
ov=2 ___ 3 ___ 7 ___12 __ 18 ___25 ___33 __...
ov=3 ___ 3 ___10 __ 22 __ 18 ___25 ___33 __...
ov= ...[text shortened]... v=7 ___ 3 ___22 __ 92 __288 __750__1716__...
But it's still very awkward to edit this!
ARR I have just spotted the pattern right now!
Providing you just ignore the v and ov values and don't try and see a pattern in terms of v and ov, it is glaringly obvious right in front of your eyes!!!
Did anyone spot it?
If you haven't spotted it yet, just completely forget about the ov and v numbers because they are a red herring, just see if you can see it in terms NOT in any way related to the row and column numbers before reading the rest of this post and look for a very SIMPLE pattern!
OK, if you still haven't spotted it and give up:
First just unconditionally accept that the first column has number 3 in all its cells (a "cell" is an entry with a unique row and column number coordinate in the table ) and the first row, except for its first cell, has number 1 in all its cells.
In other words, just accept that the first row and column has values:
3 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ 1 _ ....
OK, now simply apply the rule that, for each of the cells not in the first column or the first row, the number it contains is simply the sum of the number in the cell immediately to the left of it and the number in the cell immediately above it; -it really was that simple! but I guess I wasn't the only one not to spot it until just now!
The pattern isn't exactly the kind of pattern I was looking for but, for my purposes, this will certainly do!
I have now discovered a totally new kind of probability and some of its important properties and plan to put that in my book that I will eventually get published.
One thing I will have a go at in due course is to see if I can work out WHY that mathematical function produces that pattern! Because, for now at least, its a complete mystery to me!
It wouldn't really matter for my work if I couldn't explain why, but still nice to explain why I think.