Idea for OEIS plotting: Quasilog plots

[David W. Wilson]

I was thinking of an improvement to the OEIS logarithmic graphs.

 

I acknowledge the importance of the log graphs. For example, if a sequence
has extremely fast growth, the regular plot is unrevealing, as most of the
elements are plotted at 0. An example is A000142, where the normal plot is
pretty much useless, while the log plot nicely reveals smooth mildly
superexponential growth.

 

While the current log plots are very useful, I do have a couple of peeves.

 

My minor peeve is that the log plots are inconsistent. Either log(|a(n)|+1)
or log(|a(n)|) and sometimes just n is plotted depending on the sequence's
range and whether or not it includes 0. For instance, compare A000142,
A000179 and A008683.

 

My second peeve is a little more peevish. Specifically, when a sequence
includes both large positive and negative values, these values are both
plotted in the positive direction. This is particularly troublesome when
large positive and negative values are closely interleaved. For instance,
the log plots of A039834 and A053495 are rather deceiving.

 

I have a suggestion for improving the log plots.

 

I suggest that instead of log10(x), we do log plots with the quasi-log10
function

 

    [1] qlog10(x) = log((x+sqrt(x^2+4))/2)/log(10).

 

qlog10 satisfies

 

    [2] qlog10(10^x-10^(-x)) = x.

 

This is, q is the inverse of x -> 10^x-10^(-x). Thus q is a nice smooth
increasing R->R function.

 

 From [2], we obtain some nice graphical properties of qlog10.

 

    [3] qlog10(-x) = -qlog10(x).

 

qlog10 is an odd function. positive, negative, and zero a(n) will map to
positive, negative and zero y-coordinates, respectively. [3] also allows us
to implement

 

    [4] qlog10(x) = [1] if x >= 0; -qlog10(-x) if x < 0

 

which avoids unstable calculations of [1] on large negative arguments.

 

We also have

 

    [5] For x >= 1, 0 < qlog10(x)-log10(x) < log10(2)/x^2.

 

In other words, qlog10(x) is close enough to log10(x) for graphing purposes.
Specifically, qlog10(10) = 1.00428..., a perturbation of less that 1/2 %,
which means that 10^n will visually map very closely to y-axis tick mark n
for n >= 1. Even qlog10(1) = .20898+ is not too far from 0. qlog10(x) is
definitely more accurate than log(x)+1, currently used for many sequences.

 

To summarize, plotting qlog10(n) gives a plot that preserves order and sign
and closely approximates sgn(n)*log10(n) fairly well for all but the
smallest n, and which can be used to plot any sequence regardless of its
range.

 

For someone with plotting capabilities, it might be instructive to create a
graph which compares log10(x), -log10(-x) and qlog10(x).

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