Q: How can I find the next number in this series? Are "complete this series" problems well defined?

S: Since there are infinitely many formulas that will fit any finite series, many people object that such problems have no good answer. But isn't this a special case of the general observation that theory is underdetermined by experience (in other words, that there are a lot of world views that are consistent with all the facts that we know)? And if so, doesn't this objection really apply to all puzzles? Isn't it just more obvious in the case of series puzzles?

As a long-time observer of rec.puzzles nit-picking, I have never seen a puzzle answer that could not be challenged. The list of assumptions made in solving any puzzle is neverending. Luckily, most of us share all or nearly all of these assumptions, so that we can agree on an answer when we see it.

All of this has a lot to do with topics such as computational complexity, algorithmic compressibility, Church's thesis, intelligence, and life.

However, if you really have a series you need to find, you may be in luck. The most comprehensive collection of series in the world is

- available via email. It is
The On-Line Encyclopedia of Integer Sequences

N. J. A. Sloane

AT&T Bell Labs, Murray Hill, New Jersey

with the assistance of Simon Plouffe

Universite' du Quebec a' Montreal

To look up a sequence in the Encyclopedia, send mail to

sequences@research.att.com

containing a line of the form

lookup 4 9 16 25 36

for each sequence (up to a limit of 5) that you would like looked up.

The reply will report all sequences found in the Encyclopedia (up to a limit of 7) that match: your sequence, your sequence with 1 subtracted from each term, your sequence with 1 added to each term.

If there are too many matches, of course you should try again giving more terms!

Notation: %I = identification line: Annnn = absolute catalogue number of sequence,

Nnnnn = number (if any) in "Handbook of Integer Sequences" (1973)

%S, %T = beginning of sequence
%N = name, %R = references, %Y = cross-references, %A = authority,
%F = formula (if not included in %N line),
%O = offset = __a,b__?: a is subscript of first entry, b gives the

position of the first entry >= 2.

References to journals give volume, page, year.

New sequences, comments, corrections, extensions, etc., accompanied whenever possible by references, should be sent to: N. J. A. Sloane, ATT Bell Labs, Room 2C-376, 600 Mountain Ave, Murray Hill, NJ 07974, USA. email: njas@research.att.com, fax: 908 582 3340, voice: 908 582 2005

Ideally, new sequences and other contributions should follow the standard format, which is illustrated by:

%I A1034 N2311 %S A1034 60,168,360,504,660,1092,2448,2520,3420,4080,5616,6048,6072,7800, %T A1034 7920,9828,12180,14880,20160,25308,25920,29120,32736,34440,39732,51888 %N A1034 Orders of non-cyclic simple groups. %R A1034 DI58 309. LE70 137. ATLAS. %O A1034 1,1 %A A1034 njas

Of course the Annnn number has to be assigned by me, so use A0000 if sending a new sequence. The %S and %T lines are restricted to a total of 144 characters (digits and commas only, no blanks) The %N line may contain mathematical formulae, which are presently set in troff (though tex or latex are also acceptable). %R If a new sequence is from a preprint, please send me a copy (hard or soft), also all available publication details. If from a journal or book, please give all details, including page numbers. %A Use your email address as the authority. E.g. %A A0000 mary@this.that.edu %O Described above, but

- here is an example
%S A2885 1,1,0,1,0,0,1,2,0,4,7,0,12,8,0,80,84,0,820 %N A2885 Cyclic Steiner triple systems of order $2n+1$. %R A2885 GU70 504. %O A2885 0,8

The "0" means that the first entry gives the number of cyclic Steiner systems of order 2n+1 when n=0. The 8 means that the 8-th term is the first that is >= 1 (this determines the place of the sequence in the lexicographic order in the table).