Sequence conjectures

[Gene Smith]

Here's a sequence definition and a conjecture about it, which makes me ask,
what do you do with such a thing (beyond proving the conjecture, of course,
and even then there's a problem.)

Let f_1 = z, and f_n = f_(n-1)^2 + z. Now define the nth Mandelbrot curve,
M_n, as the curve in the real plane defined by the complex equation |f_n(z)|
= 1. That is, substututing both x+iy for z, and x-iy for z, and multiplying,
obtain a polynomial in x and y G_n of degree 2^{n+1), and then the
Mandelbrot curve is G_n = 1. The reason for bringing it up is that it
converges to the boundry of the Mandelbrot set.

The genus of a nonsingular plane curve of degree 2^(n+1) is
(2^n-1)(2^(n+1)-1). I'm conjecuring, based on the first five values,that the
correct genus for the Mandelbrot curve is (2^n-1)^2, for a sequence
0,1,9,49,225, ... which isn't in the sequence directory, but could
be--except that its pretty short!

However, I have a fix in mind for it being short.I think I can prove this
(the curve seems to have an ordinary
2^n-fold point, and no other singularities.) In that case the sequence could
go into the sequence database, but now--do you rely on my proof? What am I
supposed to do here? This *does* strike me as an interesting sequence!
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