On 2014-05-05 07:01, Fred Lunnon wrote:

Very nice pics in the book!

Thanks!

(Neckties? T-shirts? Bedspreads? Wallpaper? Um, maybe not wallpaper ...)

WFL

I am hoping the silence from the rest of you means at least a few took a crack at the sequence. 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. "Shortlly" 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