[seqfan] Chewing an old bone

[David Wilson]

Some time ago, I made a suggestion for improving the OEIS graphs. At the 
time, most of the seqfans
seemed  favorable to the idea, but it eventually dropped off the radar. 
I still think it's a good
idea, so I'm resurrecting it to see if it might fly.

(In the following, log means log_10).

In the current OEIS sequence scatter plots:

     - For small valued sequences (e.g. A001511), a(n) is plotted.
     - For large positive valued sequences (e.g. A000041), log(a(n)+1) 
is plotted.
     - For large valued sequences with negtive values, (e.g, A103718), 
log(|a(n)|+1) is plotted.

I think an improvement can be made to the log plots. Specifically, 
log(|a(n)|+1) maps both a(n) = x
and a(n) = -x to the same value above the x-axis, which makes it 
impossible to distinguish positive
and negative sequence values in a log scatter plot. To see the effect, 
look at the graphs of A103718.
The sequence values alternate between positive and negative values, 
which is apparent from the pin plot,
but is not deducible from the scatter plot.

My suggestion to improve this situation is as follows: For large-valued 
sequences, instead of plotting
log(a(n)+1) or log(|a(n)|+1), plot

     qlog(n) = arcsinh(a(n)/2)/(ln 10)

(qlog stands for quasilog). You can think of qlog as a sign-preserving 
log. From a graphing standpoint,
a qlog plot has many of the desirable features of a log plot, most 
notably, it shrinks very large values
into a reasonable plot range. qlog has the additional desirable features:

     - qlog is defined and continuous on domain R:
         - no tweaks needed to plot values <= 0 as with log.
         - suitable for plotting integer sequences, since Z is a subset of R
         - also suitable for plotting real functions if the need arises
     - qlog is monotonically increasing:
         - increases and decreases in sequence values are preserved in 
the graph.
         - ditto for sequences that include negative values.
     - qlog is an odd function:
         - signs of values are preserved in the plot.
         - positive values plot above, negative values below, zero 
values on the x-axis.
         - values of equal maginitude plot at equal distance from x-axis.
     - |qlog(n)| is asymptotic to |log(n)|.
         - for large positive n, qlog(n) is very close to log(n).
         - for large negative n, qlog(n) is very close to -log(-n).
         - for |n| >= 5, the relative difference is < 2%, and shrinks 
quickly with increasing n.
         - for positive valued sequences, qlog plots are visually 
equivalent to log plots.

Perhaps the OEIS might consider replacing its log plots with qlog plots.