t:= 2 ;  # to get about t times 1000 terms 
h:= proc() false end: v:= proc() false end: 
d:= proc() false end: u:= proc() false end:  
h(0), v(0), d(0), u(0):=true$4: 
xy:=[0, 0]: l:=0: n:=0: dd:= 1:  
dir:= [[1, 0], [0, 1], [-1, 0], [0, -1]]:  
for len to 1430*t do
  for s to 2 do
    for hh to len do
      n:= n+1; xy:= xy+dir[dd]; x, y:=xy[];
      if h(y) or v(x) or d(x+y) or u(x-y) then
        if (dd::odd) and h(y) or (dd::even) and v(x) 
          then xy:= xy+dir[dd]*(len-hh);
               n:= n+(len-hh); break
        fi
    else h(y), v(x), d(x+y), u(x-y):=true$4; 
         l:=l, n;
      fi      
    od;
    dd:= (dd mod 4)+1 
  od;
  if (len mod 710)=1 then print(len, nops([l])) fi   
od:
a:= i-> l[i+1]: 
nn:= nops([l]);         # this many terms we got
seq(a(n), n=0..60);     # first few terms