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%I #21 Dec 24 2014 23:08:01
%S 4,5,5,6,2,9,7,0,6,2,0,3,6,3,7,9,9,2,1,7,9,5,7,5,3,5,4,0,6,6,0,0,3,7,
%T 9,5,0,3,4,2,8,5,5,3,9,2,2,6,0,8,2,0,0,4,4,7,0,0,2,9,6,8,8,3,4
%N Decimal expansion of the maximum value of r such that the non-integer sequence, f(n+1) = exp(f(n)/f(n-1)) - r, grows monotonically, with f(0) = 1 and f(1) = 1.
%C Positive values of r dampen the fast growth of f(n).
%C As r approaches the value given here ("rmax") from below the monotonic growth of f(n) is dampened maximally without driving f(n) into decline.
%C When r is greater than rmax f(n) will decline to values less than 1, at some n before resuming growth again, followed by computational overflow. For r infinitesimally greater than rmax this decline can be postponed indefinitely.
%C For r infinitesimally less than rmax the growth of f(n) eventually accelerates dramatically to computational overflow, without ever declining.
%C The values of f(n) converge for r =~ rmax, at n sufficiently less than the point of decline or acceleration to overflow.
%C The initial 12 values of f(n) for r =~ rmax are given here to 7 or more digits: 1, 1, 2.262652, 9.152908, 56.66769, 487.9872, 5493.391, 77438.36, 1324625.11, 26843654.48, 632439781.27, 17062045031.59.
%C See Mathematica code below for more digits, and additional f(n) values, with characterization of their relative convergence at different n values.
%e 0.45562970620363799217957535406600379503428553922608200447002968834...
%t rmax = .45562970620363799217957535406600379503428553922608200447002968834
%t RecurrenceTable [ {a[n + 1] == Exp[ a[n]/a[n - 1]] - rmax, a[0] == 1,
%t a[1] == 1}, a, {n, 1, 48}, WorkingPrecision -> 100]
%t (* These two input lines generate monotonic growth in f(n) up to n = 41 and f(41) =~ 8 * 10^67. Overflow on f(n) happens at n = 42 (on my machine). If the last (65th) digit in the rmax assignment here is changed from 4 to 5, and thus exceeding rmax, then f(42) < f(41). This causes f(43) and f(44) =~ 1- rmax since the decline in f(n) is so great. Now overflow happens at n= 48.
%t Some additional digits in rmax may be obtained through more "trial and error" and increased working precision. Or there may be other, better methods to do so.
%t The convergence of f(n) values is characterized by comparing the stability of f(n) values for the small change, 10^(-65), in rmax as used above. f(40) =~ 4.28 * 10^65 remains stable to only these 3 digits with that small change in rmax assignment, but f(20) =~ 1.2177 * 10^25 remains stable to 42 digits (vs. the 65 digits of rmax). *)
%K nonn,cons
%O 0,1
%A _Richard R. Forberg_, Oct 16 2014
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