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Topic: algebraic numbers
Replies: 17   Last Post: Jan 8, 2010 4:16 AM

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DrMajorBob

Posts: 1,448
Registered: 11/3/08
Re: algebraic numbers
Posted: Jan 2, 2010 5:05 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

When I clicked on the link below, the search field was already filled with
the sequence

target = {1, 2, 3, 6, 11, 23, 47, 106, 235};

Searching yielded "A000055 Number of trees with n unlabeled nodes."

I tried a few Mathematica functions on it:

FindLinearRecurrence@target

FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}]

(fail)

FindSequenceFunction@target

FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}]

(fail)

f[x_] = InterpolatingPolynomial[target, x]

1 + (1 + (1/
3 + (-(1/
12) + (7/
120 + (-(1/
60) + (1/144 - (41 (-8 + x))/20160) (-7 + x)) (-6 +
x)) (-5 + x)) (-4 + x)) (-3 + x) (-2 + x)) (-1 + x)

and now the next term:

Array[f, 1 + Length@target]

{1, 2, 3, 6, 11, 23, 47, 106, 235, 322}

But, unsurprisingly, the next term in A000055 is 551, not 322.

A000055 actually starts with another three 1s, but that doesn't change
things much:


target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235};

FindLinearRecurrence@target

FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}]

(fail)

FindSequenceFunction@target

FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}]

(fail)

f[x_] = InterpolatingPolynomial[target, x]

1 + (1/24 + (-(1/
40) + (1/
90 + (-(1/
280) + (1/
1008 + (-(43/
181440) + (191/3628800 - (437 (-11 + x))/
39916800) (-10 + x)) (-9 + x)) (-8 + x)) (-7 +
x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) (-2 + x) (-1 +
x)

Array[f, 1 + Length@target]

{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502}

So I ask you, from the data alone: what's the next term?

If one had the Encyclopedia of Integer Sequences handy, those SAT
questions could be interesting. But they'd still be nonsense.

Bobby

On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <jpd@noqsi.com> wrote:

> On Dec 31, 1:16 am, DrMajorBob <btre...@austin.rr.com> wrote:
>

>> This is a little like those idiotic SAT and GRE questions that ask
>> "What's
>> the next number in the following series?"... where any number will do.
>> Test writers don't seem to know there's an interpolating polynomial (for
>> instance) to fit the given series with ANY next element.

>
> Explanations in terms of epicycles may be mathematically adequate in a
> narrow sense, but an explanation in terms of a single principle
> applied repeatedly is to be preferred in science. The ability to
> recognize such a principle is important.
>
> And my mathematical logician son (who's looking over my shoulder)
> directed me to http://www.research.att.com/~njas/sequences/ for
> research on this topic. When he encounters such a sequence in his
> research, he finds that knowledge of a simple genesis for the sequence
> can lead to further insight.
>



--
DrMajorBob@yahoo.com




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