Sequences to analyze

[Richard Guy]

I hesitated to answer this, as I had only negative
things to say, but I also find I have something
positive, so some of you may like to read on.

Negative things first:  David is so experienced in
this that I hesitated to ask, have you tried differences,
sequence fans, number walls, superseeker, ... ?  and
`where did they come from?'  This last raises a rather
metaphysical question.  How is a sequence defined?
If it arises in some obscure context, it may never
(what, never?) occur in any other, so that is its
definition.  Fibonacci numbers are defined by ...
Catalan numbers are defined by ...  But you get the
picture.

Now for something positive.  I have a great raft of
(almost certainly) new sequences (but I must admit to
not having done any of the things mentioned above).
They arise from investigating `Put-or-Take" games,
but you don't need to be interested in them
(Epstein's Game is the prototype, see WW pp.484-486,
501-503), though if you're not, you miss unbounded
numbers of sequences of nim-values, remotenesses,
suspense numbers, ...

Much more basic: start with a heap of 1, and add
the largest square which is not greater than the
present heap size:

1,2,3,4,8,12,21,37,73,137,258,514,998,1959,3895,7739,...

or, with triangular numbers:

1,2,3,6,12,22,43,79,157,310,610,1205,2381,4727,9383,...

(E&OE, done mentally, so please check).

So, another metaphysical question: when is a sequence
interesting?  I think that Neil is very generous here,
and takes a lexicographic approach, recording everything.
Some remarkable relations have been discovered from this.

I ask this last question, because, if these sequences are
to be added to the database, what can be said about them?

And what others do we add?  cubes ?  squares + 1 ?
Fibonacci numbers + 1  (stretching it a bit, I hear you
cry, but in fact that's Mike Guy's game of Fibulations,
the only non-trivial Put-or-Take game for which we have
a complete analysis)  triangles + 1 ?  numbers 2^k - 1 ?
Or any other sequence in the database!!         R.

PS:  Just try it with odd numbers.              R.

On Fri, 4 May 2001, David W. Wilson wrote:

> Here are a couple of sequences for analysis:
> 
> They have not been submitted to the EIS yet; I will do so soon.
> I have been able to crack some similar sequences, however, these
> resist my efforts.
> 
> 1,4,12,44,144,528,1808,6676,23536,87568,315136,1180680,4314560,16263896,
> 60138816,227899484,850600944,3238194560,12177384544,46542879384,
> 176110444736,675431779856,2568878867200,9882068082112,37747540858240
> 
> 1,4,14,54,200,776,2940,11466,43980,172170,665544,2612764,10154144,
> 39949000,155864280,614260062,2403739140,9486263092,37209147800,
> 147012850512,577741491404,2284848892872,8993216244896,35595538140656