login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A248926 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. 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 (list; constant; graph; refs; listen; history; text; internal format)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 16 12:30 EDT 2019. Contains 327098 sequences. (Running on oeis4.)