1.  rep(1,n)

2.  1.0, just like p_31 in the 3-H case.   

3.  need P^2

twodays <- function(v,w,r) {
   tm <- matrix(rep(0,r^2),nrow=r)
   for (i in 3:r) {
      tm[i,i-1] <- v
      tm[i,i-2] <- w
   }
   tm[2,1] <- v
   tm[2,r] <- w
   tm[1,r] <- 1
   tm %*% tm
}

print(twodays(0.6,0.4,5))

4.  Long-run proportion of 1s is pi_1 = 0.2857143 from the book.  Of the
times we have a 1, a proportion p_1,2 = 0.5 will be followed by a 2.  So
the long-run proportion of 12s is about 0.1429.