

A248926


Decimal expansion of the maximum value of r such that the noninteger sequence, f(n+1) = exp(f(n)/f(n1))  r, grows monotonically, with f(0) = 1 and f(1) = 1.


0



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, 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
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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.


LINKS

Table of n, a(n) for n=0..64.


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

Sequence in context: A046780 A075129 A021691 * A107575 A205677 A178400
Adjacent sequences: A248923 A248924 A248925 * A248927 A248928 A248929


KEYWORD

nonn,cons


AUTHOR

Richard R. Forberg, Oct 16 2014


STATUS

approved



