# [seqfan] Connection between Galton-Watson branching process and Julia sets?

Alonso Del Arte alonso.delarte at gmail.com
Thu Jun 2 04:41:39 CEST 2016

```Given z_0 = i = sqrt(-1), z_n = z_(n - 1)^2 + 1/4, we get a sequence of all
real numbers (aside from z_0) and all positive numbers (aside from z_0 and
z_1): i, -3/4, 13/16, 233/256, 70673/65536, etc. Obviously the denominators
are powers of 4. My first thought for the numerators was Fibonacci numbers,
but the coincidence is broken by 70673. Just the first four real numerators
are enough to bring up A015701, which involves iteration in the
Galton-Watson branching process.

The mention of iteration suggests I got the right match, but I had never
heard of Galton-Watson before. From what I've read in the second Google
result http://galton.uchicago.edu/~lalley/Courses/312/Branching.pdf , it
makes sense that the denominators would be powers of 4 as well. But the
OEIS entry says nothing about fractals or at least fractions. I've been
scrutinizing the Julia sets for quadratic functions with c purely real.

Any thoughts?

Al

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Alonso del Arte
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