2D self-trapping walks

[all at abouthugo.de]

Hi SeqFans,

when trying to understand my results of counting the self-avoiding walks
terminated by self-trapping after n steps, I made the
observation stated as a comment in A076874, that the number of steps,
for which there are only two choices for the direction
(constrained steps) in a trapped walk of lenght n never exceeds
n-floor(sqrt(4*n+1))-2. I checked that until n=23, but I have no
proof. A illustration of such walks is at
http://www.randomwalk.de/stw2/conswalk.pdf
The idea for this relation came from counting the number of constrained
steps in "spiral shaped" paths as illustrated in
http://www.randomwalk.de/stw2/ex2asymp.pdf (ignore the erroneous formula
in the second graph).
Any ideas how to prove this?

Thanks
Hugo Pfoertner
http://www.pfoertner.org/