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

**Follow-Ups**:**Re: algebraic numbers***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>