Re: Part specification... is neither an integer nor a list of integers

*To*: mathgroup at smc.vnet.net*Subject*: [mg109702] Re: Part specification... is neither an integer nor a list of integers*From*: Albert Retey <awnl at gmx-topmail.de>*Date*: Wed, 12 May 2010 07:35:24 -0400 (EDT)*References*: <hsbbft$jmd$1@smc.vnet.net>

Am 11.05.2010 12:28, schrieb Chandler May: > Hi Mathematica sages, > > I want to implement a recursive function on the natural numbers: > > g(n) = n - g(g(n-1)) > g(0) = 0 > > First I tried the following in Mathematica. > > g[0] := 0 > g[n_] := n - g[g[n-1]] > > This worked, but it was much too slow. In hopes of reducing the > number computations, I thought I would make a function gseq[n_] to > generate the sequence of successive values of g(n) like so: > > gseq[0] := {0} > gseq[n_] := With[{s=gseq[n-1]}, Append[s, n - s[[Last[s]]]]] > > However, when I ask for gseq[n] for n > 1, Mathematica complains that > the "Part specification... is neither an integer nor a list of > integers", like the first line here > <http://reference.wolfram.com/mathematica/ref/message/General/pspec.html> > (sorry, I don't have Mathematica in front of me at the moment). > gseq[1] gives me something like {0, 1 - List}. > > What exactly is going wrong, and how do I mend it? Also, in the With > construct, will gseq[n-1] be evaluated once and stored in s, or will > every instance of s be replaced by a call to gseq[n-1] (so that > gseq[n-1] is wastefully evaluated three times per call to gseq[n])? > If gseq[n-1] will be evaluated more than once (per call to gseq[n]), > is there a way to change the code so that it won't be? If there's a > better way to efficiently implement g(n) altogether, please share (but > please don't reveal any mathematical properties about the particular > function g(n)--don't spoil my fun). I haven't looked at the details of your problem, but the actual problem is perfect for the technique of memoization, which is what I think you try to implement in a much more complicated way. This works in a matter of, well milliseconds: g[0] = 0 g[n_] := g[n] = n - g[g[n - 1]]; Note that whatever recursive approach you want to use, you need to increase $RecursionLimit, otherwise the code will produce errormessages and huge results... In[4]:= Block[{$RecursionLimit = 100000}, Timing[g[10000]]] Out[4]= {0.078, 6180} For very large values, there seem to be other limits for the recursion depth which make kernel quit on my machine, but I think that is probably true for every recursive approach... hth, albert