2-dimensional version of Fibonacci numbers

[Marc LeBrun]

[adding seqfan for reference, Neil to get his opinion]

 >=Mitchell Harris
 > What would you think about updating the table so that it includes the 
0th cases?
 > That is, IS(n,0) = IS(0,m) = 1:
 > 1 1 1  1  1
 > 1 2 3  5  8
 > 1 3 7  17 41
 > 1 5 17 63...
 > 1 8 41 ...
 > This just makes me feel more comfortable seeing the combinatorial base
 > case (there's exactly 1 way to have an independent set on P_n x P_0: since
 > there are no vertices, there is exactly one way to choose the vertices
 > that are either in the independent set or not (and that is to choose none
 > of them)).

I too am very inclined in general to cover such 0 cases.  However for 
tables lately I'm leaning toward just adding a comment such as T(n,0)=T(0,n)=1.

How does that sound here?  If you agree, feel free to go ahead and submit a 
comment.


For the record, my reasons for this practice include:

A. For ordinary sequences, including A(0) only adds one more term, but for 
tables the T(n,0)/T(0,n) borders dilute the sequence with quite a few more 
"trivial" values, for example in multiplication tables where 0 is nilpotent.

B. I think superseeker will now find the table even when you query it in 
the diluted form.  (Neil?)

C. The A(0,0) value in particular can be exceptional or controversial 
(consider T(i,j)=i^j say) and so not convenient to include (if it *is* 
well-defined yet exceptional then that actually argues *for* including the 
borders to capture this information).

D. The interpretation or meaning of the border cases can be 
problematic.  In this example the original description was in terms of 
matrices, not sets of vertices or whatever.  While I'm sure you can 
probably define nX0 and 0Xn matrices somehow, it's not intuitively obvious 
what they are (eg how would you tell them apart?<;-).  In fact sometimes 
equivalent descriptions which are isomorphic and well-defined for i,j>0 
diverge or break on the boundaries.

E. Inserting the border cases can sometimes discombobulate pre-existing 
formulas, programs, and transforms, because the mapping between indices 
(n)<->(i,j) is different.


Generally if there's already some existing convention in play I try to 
consistently stick to it, whether or not I prefer it, because changing 
things can lead to confusion and errors (eg in this case I would have been 
much happier if the original definition would have been in terms of 
adjacent 1s instead of 0s, because that's how the Zeckendorf / Fibbinary 
stuff is usually defined!)