[math-fun] What is the period of this integer sequence?

4 May 2014

      0, 1, 7, -2, -8, 3, 14, -4, -16, 5, 21, -6, -24, 7, 28, -7, -25, 7,
27, -7, -26, 7, 26, -6, -19, 5, 18, -4, -12, 3, 10, -2, -5, 1, 2, 0,
2, -1, -6, 3, 16, -5, -22, 7, 30, -8, -30, 8, 30, -7, -23, 6, 22, -5,
-16, 4, 14, -3, -9, 2, 6, -1, -2, 1, 5, -1, -3, 1, 4, -1, ...

a[0] = 0; a[1] = 1;
a[k_?OddQ] := a[k] = a[k - 2] - Floor[9*a[k - 1]/17];
a[k_?EvenQ] := a[k] = a[k - 2] + Floor[15*a[k - 1]/2]

This is the "Minsky Stock Index" that Corey and Julian ran fruitlessly
from a[-10^14] to a[10^14] a few years ago.  "If you haven't looked at a
 problem in the last few years, you haven't looked at it." --Ed Pegg, Jr.
There is a slight chance that the period is infinite.
(When it reached 18 trillion, we exclaimed AT&T!)

The above recursive definition crashes Mathematica in under a million,
even with memoizing.  Instead use the iteration
In[115]:= NestList[Function[xy, {#, xy[[2]] + Floor[15*#/2]} &
      [xy[[1]] - Floor[9*xy[[2]]/17]]], {1, 0}, 35]
e.g., for {a[-1],a[0}, {a[1],a[2]},...{a[69],a[70]}
Out[115]= {{1, 0}, {1, 7}, ..., {-1, -4}}

Background:  http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition
Unlike Collatz or twin prime searching, this is a very specific question
about
very specific quantities, rather than a question about the infinitude of
integers.
--rwg

Bill Gosper

Fred Lunnon

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