First we see that given a fixed corner as apex, there are 4 classes of corners:
1)A(1): the apex.
2)B(3): next to apex.
3)C(3): 2 steps from apex.
4)D(1): the opposite corner.
In general we have PAA(N)=PAB(N-1)=1/3 PAA(N-2)+ 2/3 PAC(N-2)
PAC(N)=1/3 PAD(N-1) + 2/3 PAB(N-1) = 1/3 PAC(N-2) + 2/3(1/3 PAA(N-2)+ 2/3 PAC(N-2))
= 7/9 PAC(N-2)+ 2/9 PAA(N-2)
Given PAA(1)=PAC(1)=0, all PAA(N)=0 for N= odd.
Now let's work out the N=even number case
Therefore working with this pair:
PAA(2N)=1/3 PAA(2N-2)+ 2/3 PAC(2N-2)
PAC(2N)= 2/9 PAA(2N-2) + 7/9 PAC(2N-2)
Solve the eigenvalue problem of
1/3, 2/3
2/9, 7/9
or
lam^2-(10/9) lam +17/81=0... eigen value = (5+/-sqrt(8))/9
Someone finish this for me?