How to Find Patterns of Sequences

Date: 7/16/96 at 11:56:36
From: Anonymous
Subject: Find Pattern of Sequence

Hi,

Could you help me with the following number sequences:
2,3,1,2,8,9 and 6,10,15,23,31,41

Is there a method that you use to recognize and solve the sequence?
Please explain how to solve the above. Thank you very much.

P.S. Is there software available to help with this type of problem?

Date: 7/16/96 at 17:40:38
From: Doctor Anthony
Subject: Re: Find Pattern of Sequence

The short answer is that there is no unique rule to generate this 
sequence of numbers.  I could construct a polynomial of degree 7 to 
give you the numbers you list, but there can be any number of rules 
that would give the same set of numbers.  A trivial example is 2,4,6 
which could be the start of an AP with common difference 2, or the 
start of a Fibonacci series with the third term the sum of the two 
previous terms. 

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/16/96 at 11:56:36
From: Anonymous
Subject: Re: Find Pattern of Sequence

I appreciate your reply, but still do not understand. I think the 
answer to the first sequence is 5 as the next number, but am not sure. 
I have no idea how to work the second problem. Can you help me with 
both?

Date: 7/22/96 at 15:57:35
From: Doctor Ceeks
Subject: Re: Sequence/Series

Hi,

I agree very much with Doctor Anthony's response.

Any finite set of numbers can be continued in countless ways to form
a sequence.

In your problems, the hope is to find a sequence which is as simple to
understand as possible.  However, it is sometimes hard to know what is 
simplest.

In any case, many sequences have appeared in natural ways in the 
course of mathematical research, and many of these sequences are 
recorded in "Handbook of Integer Sequences" by NJA Sloane.  Dr. Sloane 
also wrote a computer program which will try to find sequences to 
which your sequences belong.  To use this program follow these steps:

Send e-mail to "sequences@research.att.com" with the message:

lookup 2 3 1 2 8 9

In a little bit, you will get e-mail from the program with any
sequences it finds and bibliographic references to where the sequences
found appear in the mathematical literature.

Have fun!

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/