Dear Sir/Madam,

I have a question related to series of numbers.

It might be related to Mathematics or Statistics or both.

It is a series which has been generated according to some formula.
Actually I need that formula which will fit into the given series and
can generate next numbers.

For example I pick one number from series ?132533? now if I will apply
some formula I will get next number i.e. ?266315? and again if I will
apply I will get next number i.e. 156070 and? 757243 and so on.

Following is series

132533
266315
156070
757243
705328
141969
253454
199338
227112
606670
930471
639455
966519
530753

 Please ask for any future details.

Regards,
Noman Jamil

1. To fit a finction one needs a series of arguments in addition to
your series of responses. For example, the arguments can be the first
14 intergers, so that you will fit a function for the following data:
 1 132533
 2 266315
 3 156070
 4 757243
 5 705328
 6 141969
 7 253454
 8 199338
 9 227112
10 606670
11 930471
12 639455
13 966519
14 530753

2. Whatever the arguments are, a 13th degree Lagrange Interpolating
Polynomial will do the job as long as the arguments are known.
Although it will fit your data exactly, you should expect a
significant (albeit computable) variation in between the nodes. (See
for example the link:
http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html)
    If you have smoothness requirements but the exact fit of your data
is not a must, then a (statistical) regression could be a better
choice. There are many methods available from which you can choose
depending on your requirements to the fit function.
-- RK

Dear raokramer,

Thanks for your reply, I have looked at
LagrangeInterpolatingPolynomial and it do need arguments and responses
of series, but in my case I wanted to get the next number to be
generated only from responses and it needs to be exact. Is there any
way to accomplish this.

Thanks ~

Now that I went through your problem desription again, I think I
understand it better, although there's a room for a better problem
description. Is this a part of a larger problem, knowing which may
help me to better understand the subject matter?

Anyway, here's how I see it now. You have 14 numbers listed in the above order:
a0, a1, , ... , a13. You are looking for a function f(x) that would
satisfy the following thirteen conditions:

f(a0)=a1;
f(a1)=a2;
f(a2)=a3;
...
f(a12)=a13.

Now, a Lagrange Interpolating Polynomial (LIP) of 12th degree will be
a correct answer unless you have additional requirements for the
function f().

Of course, a LIP of 14th or higher degree can satisfy those conditions
too. In fact, there is a multitude of the functions that would fit
your problem conditions as described.

You say "It is a series which has been generated according to some
formula". Do you or anyone else know anything about that formula? Can
it be a polynomial? How simple does it have to be? What are you trying
to acheive by finding this function? Does the next number in your
series have to be INTEGER by any chance? In fact, given *any*
arbitrary number a14, you can choose a polynomial f() in such a way so
that a14=f(13).

How many "next" numbers do you want to generate? Do they have to be
integers? Again if you need to generate, say, N next numbers a_14,
a_15, ..., a_{13+n} then just as above, you can choose a polynomial in
such a way that the first 13 conditions will be satisfied and the next
N conditions a_14=f(a13), ... , a_{13+n}=f(a_{12+n}) will be satisfied
too.

If this doesn't look like a satisfactory solution to you, then I think
a little more background is required as to the origin of your problem.

Thanks,
-- RK