On Tue, Jul 1, 2014 at 11:50 AM, Bill Gosper <billgosper@gmail.com> wrote:
I am amazed. If our sequence is a[n] := 5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, ... then 5 Abs[(3 a[n] - 2 Sqrt[-25 + a[n]^4])/(-25 + 4 a[n]^2)] is either a[n-1] or a[n+1], and changing the sign of the √ gives the other one!
{5 Abs[(3 #1 - 2 Sqrt[-25 + #1^4])/(-25 + 4 #1^2)], 5 Abs[(3 #1 + 2 Sqrt[-25 + #1^4])/(-25 + 4 #1^2)]} & /@
{5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, 312098738002194296165/128615821825334210638, 249850594047271558364480641/ 5354229862821602092291248, 5009631795998363486645417214683045/1935878334514951131830244285524398}
{{Indeterminate, \[Infinity]}, {5/2, 11285/1562}, {3344161/1494696, 41/12}, {44572169525/7118599318, 11285/1562}, {3344161/1494696, 654686219104361/178761481355556}, {312098738002194296165/128615821825334210638, 44572169525/7118599318}, {654686219104361/178761481355556, 249850594047271558364480641/5354229862821602092291248}, {312098738002194296165/128615821825334210638, 5009631795998363486645417214683045/1935878334514951131830244285524398}, {160443526614433014168714029147613242401001/50016678000996026579336936742637753055940,
249850594047271558364480641/5354229862821602092291248}}
So now we have *two* mysterious bit sequences--the signs of the Absand and √ that get us a[n+1] from a[n].
But they only amount to trits: In[61]:= MapAt[N, Reap[Nest[ Block[{L = {(5 (-3 #1 - 2 Sqrt[-25 + #1^4]))/(-25 + 4 #1^2), ( 5 (3 #1 - 2 Sqrt[-25 + #1^4]))/(-25 + 4 #1^2), ( 5 (-3 #1 + 2 Sqrt[-25 + #1^4]))/(-25 + 4 #1^2), ( 5 (3 #1 + 2 Sqrt[-25 + #1^4]))/(-25 + 4 #1^2)}}, (Sow[Position[L, #][[1, 1]]]; #) &@ Sort[L, Numerator[#1] > Numerator[#2] &][[1]]] &, 41/12, 999]], 1] // tim During evaluation of In[61]:= 3322.327528,2 Out[61]= {9.65752, {{4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, [...] 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4, 3, 3, 4, 3, 1, 4, 3, 4}}} In[62]:= FreeQ[%, 2] Out[62]= True I.e., for some reason, the 2nd sign pattern is never the way forward.
On Tue, Jul 1, 2014 at 1:49 AM, Bill Gosper <billgosper@gmail.com> wrote:
GAAAA--Rich privately suggested that the sign pattern in my 1st order, nonlinear recursion was not period 8, not even periodic. And he is right!!
How the freep did he know that? --rwg
The reason I got so exclamatory was the early (and middle)
lack of supporting evidence or motivation for his claim. --rwg
Watch the recurrence slam into reverse at n=57!
[big chop]