sequence
Gustav Melck
3, -2, 3, -3, 4, 4, -7, 4, -1, -2, 2, ...
Gustav!
Chris Cole
==> series/series.00.p <==
How can I find the next number in this series? Are "complete this
series" problems well defined?
==> series/series.00.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
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: nj...@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 ma...@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).
Here is the reply from super...@research.att.com:
From superse...@research.att.com Wed Oct 21 20:47:39 1998
From: superse...@research.att.com
To: ch...@questrel.com
Date: Wed, 21 Oct 1998 23:45:51 -0400 (EDT)
Report on [ 3,-2,3,-3,4,4,-7,4,-1,-2,2]:
Many tests are carried out, but only potentially useful information
(if any) is reported here.
Even though there are 40,000 sequences in the table now, at least
one of yours is not there! Please send it to me using
the submission form on the sequence web page
http://www.research.att.com/~njas/sequences/index.html#S
and I will (probably) add it! Include a brief description. Thanks!
TEST: APPLY VARIOUS TRANSFORMATIONS TO SEQUENCE AND LOOK IT
UP IN THE ENCYCLOPEDIA AGAIN
SUCCESS
(limited to 20 matches):
Transformation T021 gave a match with:
%I A000796 M2218 N0880
%S A000796
3,1,4,1,5,9,2,6,5,3,5,8,9,7,9,3,2,3,8,4,6,2,6,4,3,3,8,3,2,7,9,5,0,
%T A000796
2,8,8,4,1,9,7,1,6,9,3,9,9,3,7,5,1,0,5,8,2,0,9,7,4,9,4,4,5,9,2,3,0,
%U A000796
7,8,1,6,4,0,6,2,8,6,2,0,8,9,9,8,6,2,8,0,3,4,8,2,5,3,4,2,1,1,7,0,6
%N A000796 Decimal expansion of Pi.
%R A000796 MOC 16 80 1962.
%O A000796 1,1
%A A000796 njas
%K A000796 cons,nonn,nice
%e A000796
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303820...
%H A000796 <a href="http://www.lacim.uqam.ca/pi/records.html">More
digits</a>
References in R lines (if any):
[MOC] = { Mathematics of Computation} (formerly { Mathematical Tables
and Other Aids to Computation}).
List of transformations used:
T021 coefficients of Sn(z)/(1-z)
Abbreviations used in the above list of transformations:
u[j] = j-th term of the sequence
v[j] = u[j]/(j-1)!
Sn(z) = ordinary generating function
En(z) = exponential generating function
Gustav Melck
Chris Cole wrote:
> Gustav Melck wrote:
>
> > what are the next two terms in this sequence?
> >
> > 3, -2, 3, -3, 4, 4, -7, 4, -1, -2, 2, ...
> >
> > Gustav!
>
> ==> series/series.00.p <==
> How can I find the next number in this series? Are "complete this
> series" problems well defined?
Well, this problem is not a series, but not really a true "mathematical" sequence either (at least not like those that I'm used to).
In this case I use the term sequence to loosly mean a list of numbers that are somehow related to each other or to some fundamental pattern. I don't think that one would have much success with approaching
this particular problem very mathematically, but rather by thinking of possible solutions and then testing them out (I'm not sure if that sounds to vague). This is because I don't think that there is a
relatively simple general term for the sequence.
I'm not sure if I should post the solution yet, or if I should wait a little while longer. I think that I will, in any case, give a clue:
The underlying "pattern" (if you can call it that) is based on a mathematical constant. To solve this sequence, you would have to at least be able to recognise this constant. I suppose that mathematicians or
engineers would have a bit of an advantage, because they deal with this constant more than the average person that you'll meet on the street.
I have just read the above paragraph out to myself and I hope that it won't discourage anybody. It shouldn't be too difficult to solve now, I think! (I have probably given it away)
Thanks for all that other info. I will remember this when I have some other sequence I need to find. I am not too sure if it will work with my sequence, though!
Gustav!
Michael Dougherty
: what are the next two terms in this sequence?
: 3, -2, 3, -3, 4, 4, -7, 4, -1, -2, 2, ...
: Gustav!
(Spoiler follows)
3, 1
The numbers (after the first) are the differences which must be applied to
the Nth digit of pi in order to get the next digit. (3-2=1, 1+3=4, 4-3=1,
etc.)
Mike D.
il...@isgtec.com
Gustav Melck <s971...@student.up.ac.za> wrote:
> Hmmm...
>
> Chris Cole wrote:
>
> > Gustav Melck wrote:
> >
> > > what are the next two terms in this sequence?
> > >
> > > 3, -2, 3, -3, 4, 4, -7, 4, -1, -2, 2, ...
> > >
> > > Gustav!
> >
> > ==> series/series.00.p <==
> > How can I find the next number in this series? Are "complete this
> > series" problems well defined?
>
> In this case I use the term sequence to loosly mean a list of numbers that are somehow related to each other or to some fundamental pattern. I don't think that one would have much success with approaching
> this particular problem very mathematically, but rather by thinking of possible solutions and then testing them out (I'm not sure if that sounds to vague). This is because I don't think that there is a
> relatively simple general term for the sequence.
>
> I'm not sure if I should post the solution yet, or if I should wait a little while longer. I think that I will, in any case, give a clue:
> The underlying "pattern" (if you can call it that) is based on a mathematical constant. To solve this sequence, you would have to at least be able to recognise this constant. I suppose that mathematicians or
> engineers would have a bit of an advantage, because they deal with this constant more than the average person that you'll meet on the street.
>
> I have just read the above paragraph out to myself and I hope that it won't discourage anybody. It shouldn't be too difficult to solve now, I think! (I have probably given it away)
>
> Thanks for all that other info. I will remember this when I have some other sequence I need to find. I am not too sure if it will work with my sequence, though!
>
> Gustav!
>
SPOILER
Take pi = 3.141592653589793...
Starting with 0, take the differences between adjacet digits:
3-0, 1-3, 4-1, 1-4, 5-1, etc.
The next digit in the series is thus 3 (8-5).
__/\__
\ /
__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
/__ __\
||
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