[SeqFan] OEIS and mathematics hypertext.

[Antti Karttunen]

wouter meeussen wrote:

>dear seqfanners,
>
>the ideas exposed below are central to the OEIS too, I believe. Not only the body of internal (seq
>to seqs) and external (EW's mathworld.wolfram.com) links, but also C.G. Bower's expositions in
>"../~njas/sequences/transforms2.html" give the OEIS great didactic qualities. Possibly beyond its
>original intended use?
>
>Wouter.
>
This is probably more related to OEIS than Mayans' project
(Xanadu -> Tikal, Popol Vuh? .-) but I have been for some time
dreamt about a more exact implementation for the "schema"
of OEIS entry contents than the currently implemented one.

If there is anybody willing to learn XML, it might be a good
idea to create a specific OEIS-DTD as a study project.
(and get Neil enthusiastic also...)

Particularly, to emphasize the nature of OEIS as a database
of N->Z functions, the links between various sequences could be
represented much more exactly by using the transformations given in
http://www.research.att.com/~njas/sequences/transforms.html
with a special emphasis given to a correct handling of offsets,
e.g. by requiring use of such shifting transforms as LEFT,
RIGHT (which shifts right with 1 prepended), RIGHT0
(shifts right with 0 prepended), etc.


Let's try:

**E.g. the correspondences given for A000005 ("tau", number of divisors of n):
a(n) = A032741 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A032741>(n)+1 = A062011 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A062011>(n)/2 = A054519 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A054519>(n)-A054519 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A054519>(n-1) =
*           *   A006218 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006218>(n)-A006218 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A006218>(n-1) = sum(k=0,n-1,A051950 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A051950>(k)). 

might be represented with transforms as:

A032741 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A032741>(n)+1 as PLUS1(A032741), which would be a short hand for
SUM(A032741,A000012) (where A000012 is all-1 sequence)

(where SUM would be a new simple transform giving a point-wise
sum of two sequences)

A062011 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A062011>(n)/2 as COMPOSE(A004526,A062011), 

(where A004526 is 0,0,1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,
"Integers repeated", and works effectively as a function n -> n/2 in 
this case)
or maybe we could have a new transform short-hand HALVE for these
cases.

A054519 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A054519>(n)-A054519 <http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A054519>(n-1) as DIFFERENCE(A054519,COMPOSE(A054519,MINUS1(*A001477*)))

where *A001477 is *0,1,2,3,4,5,6,7,8,9,10,11, "The nonnegative integers."
and MINUS1(X) would be a short-hand for DIFFERENCE(X,A000012),
and DIFFERENCE(X,Y) woule be a short-hand for PLUS(X,NEGATE_ALL(Y))
or simply as DIFF(A054519), an existing transform which gives the first
differences of the sequence.


And the last, sum(k=0,n-1,A051950 
<http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A051950>(k))
could be represented as PSUM(A051950)
where "PSUM" is an existing transform for partial sums
(with some thought still needed to get the indices right).

You might think that this is even more baroque than now,
but such a simple, easily PARSABLE, COMPUTED and WELL-DEFINED
"functor"-notation might be used only internally, with some
of the commonest forms automatically rewritten to more
natural +, n-1, 2^n, etc constructs when the entry is rendered
for a pair of human eyes to see. If this kind of notation were
properly designed, it would offer a good starting point for
automatic checking of OEIS-data, and all the given inter-dependencies.

Also there would be need for such "set-based"
transforms as COMP(lement), UNION, and INTERSECT,
e.g. INTERSECT(A000040,Axxxxxx) for cases like
"Primes of the form Axxxxxx".

Furthermore, given also some consistent way to define the formulae for
the certain set of core-sequences from which other ones
could be derived through such transforms, on-the-fly
generation of sequence terms might eventually be easy to do.
Of course this would be best to do on the "client-side" with e.g.
a Java/Scheme-applet, to avoid hogging resources of
OEIS-server!

E.g. the consistent and OPEN (*) way could be a some fixed method
of writing generating functions for many of the combinatoric sequences,
but I have also played with a crazy idea of using
just numerical codes for the nth "primitively recursive
function" to get rid off all syntax-issues (well, of course that
rules out the Ackerman function and other rarities)
but this would probably require some very clever way to
enumerate them, and to effectively cache the values
of the intermediate functions. Note that the OEIS system
itself offers one such efficient method of caching, up to
the first hundred or such terms.
(Maybe a hybrid "primitive recursive function" enumeration
with references to A-numbers allowed would be needed.)

(* I.e. I think presenting just Maple or Mathematica program lines
is too proprietary, and prone to suffer from the library dependencies,
version changes, etc.)



Yours,

Antti Karttunen



>
>
>----- Original Message -----
>From: <BobMayans at aol.com>
>To: <math-fun at mailman.xmission.com>
>Sent: Sunday, May 30, 2004 5:37 AM
>Subject: [math-fun] mathematics hypertext
>
>
>Greetings.
>
>I have been working on the design of a large-scale hypertext for mathematics:
>a
>linked structure of Web pages of mathematical writing.  A paper on the design
>issues and proposed architecture of a mathematical hypertext, at
>
>   http://jodi.ecs.soton.ac.uk/Articles/v05/i01/Mayans/
>
>may be of interest to math-funsters.  This is a very large and long-term
>project.
>A first text, in linear algebra, is in the works.  The starting point of this
>work is the
>question: what is the ideal way to write mathematics?  I'm looking for
>thoughts,
>suggestions, and maybe collaborators in this work.
>
>Bob Mayans
>
>_______________________________________________
>math-fun mailing list
>math-fun at mailman.xmission.com
>http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
>
>
>
>
>  
>