[seqfan] questions about walks in the plane

[Neil Sloane]

An old friend (Kees Immink) asked me about the conjecture of David Scambler
in A085363.  In fact there are 4 assertions/conjectures in the OEIS of this
type: (the first is only a one-D walk)

%C A001630 Apparently for n>=2 the number of 1-D walks of length n-2 using
steps +1, +3 and -1, avoiding consecutive -1 steps. - _David Scambler_, Jul
15 2013

%C A084768 Number of directed 2-D walks of length 2n starting at (0,0) and
ending on the X-axis using steps NE, SE, NE, SW and avoiding NE followed by
SE. - _David Scambler_, Jun 24 2013

%C A085363 Apparently, the number of 2-D directed walks of semilength n
starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW
avoiding adjacent NW/SE and adjacent NE/SW. - _David Scambler_, Jun 20 2013

%C A101500 Directed 2-D walks with n steps starting at (0,0) and ending on
the X-axis using steps N,S,E,W and avoiding N followed by S. - _David
Scambler_, Jun 24 2013

I know we have several experts here - could someone help and provide proofs?

The third question is the following:
Let a(n) = the number of 2-D directed walks of semilength n starting at
(0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding
adjacent NW/SE and adjacent NE/SW
Show that this satisfies the recurrence
a(0)=1; for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

(The second and fourth questions are stated as if they are theorems, but no
proof is given.)

Neil