[seqfan] New plane filling sequences

[Brad Klee]

Hi Seqfans,

On the comments of A260482,

https://oeis.org/A260482

Gosper conjectures that all dragon triple points have pre-images of the
form: q = m / (15*2^n) for m in 0,1,2...15*2^n .

This apparently isn't true because there are also triple points with
pre-images of the form: q = m / (14*2^n) for m in 0,1,2...14*2^n .

Obviously these sets are different because the denominators contain
different prime factors.

The main confusing problem is that an apparent triple point could have
another pre-image in an expanded domain including possibly irrational
pre-images for which the Q-function isn't defined.

After a few hours of detailed convergence analysis, I somewhat-believe the
proposition that both of these denominator sequences have triple points
which are strictly triple-points. That is, I don't think that any of these
triple points have other surprising, unknown pre-images.

In order to avoid ambiguity and make the picture more simple, I propose a
weak definition of an n-point that depends on a set of values. See for
example:

https://oeis.org/draft/A261120
https://oeis.org/A261120

On this sequence, for each n I count the triple points occurring in the set
of functional values FLSN[ m / (6*7^n) ] for n in 1,2...6*7^n, with FLSN
the flowsnake Q-function. The first few terms are

2, 17, 134, 989, 7082, 50057, 351854 ...

And the functional form is determined by a simple system of recurrence
equations.

Now we can return to Gosper's conjecture. Under similar definitions, dragon
triple counts are ( with offset 1 ):

q = m / (14*2^n) : 1, 4, 11, 28, 67, 152, 335, 724...   ( A003230 ?? )

q = m / (15*2^n) : 1, 3, 7, 17, 43, 105, 247, 565...     ( A??????  )

I know that the functional form is the solution to a system of recurrence
equations, but I have yet to figure the systems / solutions for either
sequence.

>From the hours of convergence analysis, the connection to seq. A003230
seems very likely. If you use mathematica, copy Gosper's "dragun" function
and try:

ReIm[x_] := {Re[x], Im[x]}
VList[n_, m_] := ReIm at dragun[1/n/2^m #][[1]] & /@ Range[0, n 2^m]
Function[{n}, Graphics[{Line at VList[1, n + 3], Red,
    Point /@ Cases[Tally[VList[14, n]], {x_, 3} :> x]}]
  ] /@ Range[5]

Cool Picture ! !

I'm putting out the challenge:

Can anyone compute the functional form for these sequences and prove /
disprove that A003230 is also related to a sequence counting triple points?

I hope so.

Please send questions or comments.

Cheers,

Brad