I just noticed a typo in this item dated 2014-05-08 12:56: "Thus the only "restoring force" capable of pulling a[2n],a[2n+1] back to 0,1 ..." should say "back to (0,8)" Anyone who was confused by this is paying way too much attention. The item was primarily about the "Minsky Stock Index" restated toward the end. Last I heard, Tom Rokicki had chased it about 13 quadrillion steps without returning to (1,0). --rwg ----------------------------------------- [...] By way of couragement, here is a similar sequence we did track. a[-2], a[-1], a[0], a[1],... = 8, 17, 8, 0, -17, -17, 12, 21, 4, -5,... a[k_?OddQ] == a[k - 2] - Floor[d*a[k - 1]], a[k_?EvenQ] == a[k - 2] + Floor[e*a[k - 1]] where the exotic rationals d := 2450365/1120052, e := 12497/8798 arose from a hopeless attempt to completely map the periods in a subpatch of (d,e) space in http://nbickford.files.wordpress.com/2011/03/neighborhood.png . Instead of the scalars a[n], we ran the pairs {x[n],y[n]} :={a[2n+1],a[2n]}, which lie on a very eccentric ellipse, which can be circularized by {x',y'}:= {(x - (d*y)/2)/(d^2*(-(1/4) + 1/(d*e)))^(1/4), (d* Sqrt[-(1/4) + 1/(d*e)]*y)/(d^2*(-(1/4) + 1/(d*e)))^(1/4)} This is the period 51284705405843 (times 2 for a[n]) "monster" mentioned on p13. The radius plot of (x',y') vs time is http://gosper.org/51TRadiusHistory.png , where the time axis is compressed a billionfold. Note that it is much trendier than a random walk. The sudden market crash at the end took 3 CPU days. Casual observers may have missed the crucial fact that, despite the Floor functions, the a[n] sequence is exactly reversible, so it must return to 8,0,-17 or blow up. Likewise for the subject sequence and all similarly defined. "Shortly" after time 104T, the circularized radius is triple that of 51T's maximum. At such a large radius the Floors are very nearly no-ops, leaving the circularized radius very nearly constant. Thus the only "restoring force" capable of pulling a[2n],a[2n+1] back to 0,1 is that, until then, it must self-avoid. Somehow. --rwg On 5/4/14, Bill Gosper <billgosper@gmail.com> wrote: 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