
Re: algebraic numbers
Posted:
Jan 2, 2010 5:05 AM


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

