Tabular sequences

[David W. Wilson]

Eric W. Weisstein wrote:

> Hi, David.
>
> I sent this to NJAS and to seqfan this morning but it bounced back from
> seqfan for the moment, so I'll send it to you directly until I can get my
> e-mail address updated for submissions.  I haven't looked at your proposal
> in enough detail to see how it differs yet, but looks as if we're thinking
> along similar lines...
>
> Cheers,
> -E

I think that we were basically thinking along the same lines, and I think
both our ideas have merit.  I think mine is better with respect to describing
the index sequence, while yours addresses issues of table formatting that
mine does not.

In my proposal, the %O line syntax is altered to accomodate a description of
the index sequence.  This does not require a new label, like %i.

In my proposal, the index sequence is specified in a language that expresses
the index sequence in terms of simple explicit sequences, Sloane sequences,
and simple combinations thereof.  For instance, any of the following are valid
ways to describe an index sequence:

    "1, 2, 3, 5" for (1, 2, 3, 5)
    "1 to inf" for (1, 2, 3, 4, ...)
    "0 to inf by 2" for (0, 2, 4, 6, ...)
    "0 to -inf by 2" for (0, -2, -4, -6, ...)
    "1 to ?" for (1, 2, 3, 4, ..., unknown value)
    "1, 2, 3 to inf by 2" for (1, 2, 3, 5, 7, ...)
    "A000027" for (1, 2, 3, 4, 5, ...)  (but "1 to inf" is clearer)
    1, A000040 for (1, 2, 3, 5, 7, ...) (1 + primes)
     "A003056 X A002262" for ((0, 0), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2), (3, 0), ...)

Ability to use expressions like "5, 7 to inf" would avoid devoting A-numbers
to arcane index sequences.

It would be easy enough to augment the language with macros for common
index sequences like "N", "Z+", "Z" (for 0, 1, -1, 2, -2, ...), "primes", "primePowers",
"squarefrees", diagonalTable(0 to inf, 0 to inf) (for ((0, 0), (1, 0), (1, 1), (2, 0), (2, 1),
(2, 2), (3, 0), ...))  etc.

With regard to formatting 2-d tables, one might provide a matrix from Cartesian
orientation to the preferred orientation.  The default matrix would be "1 0 0 1",
indicating x increases to the right (1, 0), and y increases upward (0, 1).  The common
book-style table x increments right and y down if "1 0 0 -1".   Supposing the choice
numbers are indexed according to the choice function, "-1 -1 2 0" or "-1 -.5 1 0"
gives a Pascal triangle orientation.

Once table orientation is established, cells can be filled with values and the resulting
table centered on the page, so centering is not really an issue.