Michael Somos somos at
Sun Jul 25 20:31:59 CEST 1999

I am working on a long essay on sequences, but let me state a
few of my thoughts and opinions on the table issue.

Abstractly, any denumerable sequence can be numbered from
an origin index like one. This is intuitively clear. That
is not a problem. The problems arise from the case where
the index of a sequence element has significance. An easy
example of this is the sequence 1,4,9,16,... which is just
a(n)=n^2 if the elements are numbered sequentially from one.
If another origin is chosen, then the formula becomes more
complicated. Hence, a consequence of choosing a indexing
system is that it determines how complicated the formula is.
Another easy example is 1,1,2,3,5,8,... which is the Fibonacci
numbers. With origin one, we have a(n) is divisible by a(m)
if n is divisible by m. If another origin is chosen, this
nice property must be suitably modified and more complicated.

Now for some more complications. Suppose we have a power
series generating function  sum(x^(n^2))  where the sum is
over all integers. This power series is 1+2*x^1+2*x^4+2*x^9+...
and the sequence associated with it is [1,2,0,0,2,0,0,0,0,2,0,...]
but this is sparse with predictable index of non-zero elements.
Perhaps this should be written as [1,2,2,2,...] instead. This
is a possible choice and this is only one example of a much
more general situation where sparse sequences have predicatable
index of non-zero elements.

As a seemingly theoretical note, not something that the EIS has
to be concerned with, consider the situation with ordinals. Here
we have a situation where we not only have to deal with the
infinite sequences  [a1,a2,a3,....] but also [a1,a2,a3,...,b1]
or [a1,a2,a3,...,b1,b2,b3,...] and so on with more complicated
ordinals. These sequence are all denumerable and yet have much
different "shapes". It seems like we need not worry about this.

So far we have been concerned with linear or one dimensional
arrangements of sequence elements. The possibilities become
vastly greater in just two dimensions. There are various ways
to arrange sequences in spiral or zig-zag arrangements. Not
only that, but an arrangement can be "read" in a different
order giving another sequence. Here is a simple example:

        1   2   4   7  11 ...
        3   5   8  12 ...
        6   9  13 ...
       10  14 ...
       15 ...

This is the positive integer sequence arranged in a triangular
arrangement which zig-zags along the anti-diagonals of a square
array. Now suppose we attempt to "read" this square array by
its columns. We get the sequence [1,3,6,10,15,...,2,5,9,14,...,
4,8,13,...,7,12,...,11,...] and this is a different ordinal
arrangement. Clearly not something we want in the EIS. This
shows that we have to be careful in how we index sequences.
We even have simple variations like reading by the anti-diagonals
going up instead of down giving [1,3,2,6,5,4,10,9,8,7,...].

If the sequence is given by a natural description, for example,
a(n,k) = number of k-subsets of an n-set, then this is important
information. This is more important than whether it is displayed

          1               1   1   1   1         1
        1   1      or     1   2   3   4   or    1  1
      1   2   1           1   3   6  10         1  2  1
    1   3   3   1         1   4  10  20         1  3  3  1

In fact, it is not easy to determine the indexing of the elements
from such a display alone. The issues are surprising complicated.
I am still working on "What is a sequence, really?". The answers
are not obvious and need careful development. Shalom, Michael

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