[seqfan] Re: The Demichel errors, continued
Olivier Gerard
olivier.gerard at gmail.com
Sat Dec 17 23:46:07 CET 2011
>
> On Sat, Dec 17, 2011 at 22:03, D. S. McNeil <dsm054 at gmail.com> wrote:
> Here are the 64 sequences in a013538.txt which weren't trivially
> recognized by my code. I don't claim they're wrong, much less
> inexplicably so, only that they passed my first filter and that I'm
> pretty sure I can identify the remaining ones, so those interested in
> playing guess-the-sequence can concentrate their efforts below:
>
>
I concentrated on these seven
A013305 arccosh(e^x - sin(x))
A013306 arccosh(e^x - arcsin(x))
> A013307 arccosh(e^x - tan(x))
> A013308 arccosh(e^x - arctan(x))
> A013323 arccosh(e^x - arcsinh(x))
> A013324 arccosh(e^x - tanh(x))
> A013325 arccosh(e^x - arctanh(x))
>
If you compute all of them from the formula, you get a series of rationals,
not integers.
Patrick probably did not take sufficient precautions around that case.
After adjusting for offset, I think I have a description of the problem.
Here are the ratios of the correct sequence to the one in the OEIS.
You will notice a large number of 1 s and most signs are correct. This
corresponds to places
where an integer is close (around 10^-6 in relative value) to the true
rational member of
the sequence. I suspect that the code Patrick used had bugs with
rationals, sometimes making the division and giving the integer closest
to it, sometimes keeping the numerator or denominator or something more
crude or impredictable related to internal representations of integers and
decimals by the
arithmetic subsystem.
If you look closely to the Demichel file, you will see at places 0 s with
minus sign before them :
that's the trace of small negative rationals being reduced to their integer
part with an incorrect
algorithm.
Anyway his code did not check for rationals in the first place.
I suspect this problem is present in most of the other sequences checked by
Doug.
We can consider erroneous sequences from the Demichel list to be essentially
computational errors and not incorrect labeling of genuine artifacts, to
use Neil's comparison.
If we want to have in the OEIS sequences like arccosh(e^x - sin(x)), we
will need two entries
by item.
Olivier
ratios : computed by Mathematica / taken from the Oeis
A013305 arccosh(e^x - sin(x))
{1.,0.111111,-0.333333,-0.888889,-0.925926,0.0205512,1.01559,1.02716,1.00097,0.00826471,1.00011,1.,1.00001,1.,1.,1.,1.,0.00277008,1.,1.,1.,0.00189036,1.},
A013306 arccosh(e^x - arcsin(x))
{1.,0.,0.,0.04,0.,1.,1.01818,1.0005,1.00019,1.,1.,0.00591716,1.,1.,1.,0.00346021,1.,1.,1.,1.},
A013307 arccosh(e^x - tan(x))
{1.,0.111111,-0.0833333,1.19444,1.27894,0.0204285,1.00768,1.00116,1.00013,0.00826448,1.,1.,1.,1.,1.,1.,1.,0.00277008,1.,1.},
A013308 arccosh(e^x - arctan(x))
{1.,1.,-0.75,0.0408,1.13542,1.0047,1.00037,1.00003,1.0001,1.,1.,0.00591716,1.,1.,1.,0.00346021,1.,1.,1.,1.},
A013323 arccosh(e^x - arcsinh(x))
{1.,0.111111,-0.333333,0.0414815,1.74074,0.0204284,1.01312,1.00027,1.00042,0.00826447,1.,0.00591716,1.,1.,1.,0.00346021,1.,0.00277008,1.,1.},
A013324 arccosh(e^x - tanh(x))
{1.,1.,-0.75,1.16667,1.40625,1.01122,1.01172,1.00086,1.00007,1.00002,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.,1.},
A013325 arccosh(e^x - arctanh(x))
{1.,0.111111,-0.0833333,0.0419883,1.29707,0.020415,1.00191,1.00016,1.00002,0.00826446,1.,0.00591716,1.,1.,1.,0.00346021,1.,0.00277008,1.}
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