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Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106129] Re: [mg106080] Re: algebraic numbers
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Sat, 2 Jan 2010 05:05:46 -0500 (EST)
*References*: <hhc7a1$2o2$1@smc.vnet.net> <200912300912.EAA17052@smc.vnet.net>
*Reply-to*: drmajorbob at yahoo.com
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 at noqsi.com> wrote:
> On Dec 31, 1:16 am, DrMajorBob <btre... at 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 at yahoo.com
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