OFFSET
0,1
COMMENTS
Positive values of r dampen the fast growth of f(n).
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.
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.
For r infinitesimally less than rmax the growth of f(n) eventually accelerates dramatically to computational overflow, without ever declining.
The values of f(n) converge for r =~ rmax, at n sufficiently less than the point of decline or acceleration to overflow.
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.
See Mathematica code below for more digits, and additional f(n) values, with characterization of their relative convergence at different n values.
EXAMPLE
0.45562970620363799217957535406600379503428553922608200447002968834...
MATHEMATICA
rmax = .45562970620363799217957535406600379503428553922608200447002968834
RecurrenceTable [ {a[n + 1] == Exp[ a[n]/a[n - 1]] - rmax, a[0] == 1,
a[1] == 1}, a, {n, 1, 48}, WorkingPrecision -> 100]
(* 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.
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.
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). *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Richard R. Forberg, Oct 16 2014
STATUS
approved