[seqfan] slowest-growing "accelerating" sequence

Jon Wild wild at music.mcgill.ca
Fri Sep 13 05:21:14 CEST 2013


Hard to believe this is new--is it right?

a(n) = 1,2,5,16,68,404,3587,51747,1343181,70863530...

Seems more likely that I've calculated this wrong than that no one has 
thought of it before and put it in the oeis: a(n) is the slowest growing 
integer sequence beginning with 1 whose sequences of mth-order quotients 
are all strictly increasing, for all values of m.

(explanation by example in case it isn't clear: the 1st-order quotients 
relate successive elements of the sequence; they are 2, 2.5, 3.2, 4.25, 
...  The 2nd-order quotients are 1.25, 1.28, ... 3rd-order quotients are 
1.024, etc. These, and all successive mth-order sequences, are strictly 
increasing. So in a sense, the sequence is increasing, its rate of 
increase is increasing, the rate of its rate of increase is increasing, ad 
infinitum--and a(n) is the slowest sequence for which this is true.)

jon wild












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