This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = 1: The fixed initial move. a(2) = 7: Relative to the direction given by the initial move, there are 7 possible direction changes. The backward direction is illegal for the self-avoiding uncrossed path. Only for the right angle turn its mirror image would coincide with the turn in the opposite direction. Therefore this move would be eliminated in the unrooted walks, making a(2) > A323132(2) = 6. a(3) = 47: 2 of all 7*7 = 49 continuation moves already lead to a crossing of the first path segment. See illustrations at Pfoertner link.
# For starting point stp and list Ldir of n directions (1..8)
# construct the points of the whole path and count them.
# If there are n+1 then the path is self-avoiding.
isSelfAvoiding := proc(Ldir) local Delta, dir, ep, path;
Delta := [[1,0],[1,1],[0,1],[-1,1],[-1,0],[-1,-1],[0,-1],[1,-1]];
ep := [0,0]; path := {ep};
for dir in Ldir do
ep := ep + Delta[dir];
path := {op(path), ep};
od;
return evalb(nops(path)=nops(Ldir)+1);
end:
# Count only king tours which are self-avoiding
A272763 := proc(n) local count, T, p;
count := 0:
T := combinat[cartprod]([seq([$1..8], j=1..n)]):
while not T[finished] do
p := T[nextvalue]();
if isSelfAvoiding(p) then count := count+1; fi;
od:
return count;
end: # Rainer Rosenthal, Jan 05 2019
mo=Most@Tuples[{-1,1,0},2]; a[0]=1; a[tg_, p_: {{0, 0}}] := Block[{e, mv = Complement[Last[p] + # & /@ mo, p]}, If[tg == 1, Length@mv, Sum[a[tg - 1, Append[p, e]], {e, mv}]]]; a /@ Range[0, 7] (* Giovanni Resta, May 06 2016 *)
a(5) = 2: There are 2 walks where the king is blocked after 5 steps, because for the diagonal moves it would have to cross its previous path.
.
o 2 o o 3 o
/ \ / \
/ \ / \
/ \ / \
3 5 1 4 - - - 5 2
| / / /
| / / /
| / / /
4 S o S - - - 1 o
See illustrations at Pfoertner link.
For n=2, the a(2)=15 paths are: . . 0 . . 0 . . 0 . . 0 2 . 0 . . . | | | |/ \ . 1 . . 1 . . 1-2 . 1 . . 2-1 . . | \ . 2 . . . 2 . . . . . . . . . . . . 0 . . 0 . . 0 . . 0 . . 0 . 2 . \ \ \ \ \ / . . 1 . . 1 . . 1 . . 1-2 . 1 . . / | \ . 2 . . . 2 . . . 2 . . . . . . . . 0 2 . 0-1 . 0-1 . 0-1 . 0-1-2 . \| / | \ . . 1 . 2 . . . 2 . . . 2 . . . . . . . . . . . . . . . . . . . .
(see GitHubGist link)
next[x_]:=Map[x + #&, Tuples[{-1, 0, 1}, 2]]
valid[s_]:=Select[next[s[[-1]]], 0<=#[[1]] && 0<=#[[2]] && FreeQ[s,#] &]
nextpath[p_]:=Outer[Append,{p},valid[p],1]
iterate[p_]:=Flatten[Map[nextpath, p], 2]
Table[Length[Nest[iterate, {{{0,0}}}, n-1]], {n,1,7}]
a(3) = 5. The second square has coordinates (0,1) and the sum of the first two numbers is 1 + 2 = 3, which is prime. Therefore, to move as far away from the origin as possible, a step to (1,2) is taken, which has a square distance of 5 units from the origin. Note that a step to (-1,2) could have also been taken and would lead to the same walk by symmetry. a(6) = 18 as a(5) is at coordinates (3,4) and the sum of the last two square numbers is 4 + 5 = 9, which is composite. Therefore, to step to a square as close as possible to the origin, a step to (3,3) is taken, which has a square distance of 18 units from the origin. a(9) = 41 as a(8) is at coordinates (5,5) and the sum of the last two square numbers is 7 + 8 = 15, which is composite. Two squares as close as possible to the origin are available, (4,5) and (5,4), both of which have a square distance from the origin of 41 units. Since (4,5) has a square distance of 32 units from the square numbered 2, and (5,4) has a square distance of 34 units from 2, the former is chosen.
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