A189722 Number of self-avoiding walks of length n on square lattice such that at each point the angle turns 90 degrees (the first turn is assumed to be to the left - otherwise the number must be doubled).

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 141, 226, 362, 580, 921, 1468, 2344, 3740, 5922, 9413, 14978, 23829, 37686, 59770, 94882, 150606, 237947, 376784, 597063, 946086, 1493497, 2361970, 3737699, 5914635, 9330438, 14741315, 23301716, 36833270, 58071568

Offset: 2

Author

Dan Dima and Stephen C. Locke, Apr 25-26 2011

Keywords

• nonn
• walk

Comments

The number of snakes composed of n identical segments such that the snake starts with a left turn and the other (n-2) joints are bent at 90-degree angles, either to the left or to the right, in such a way that the snake does not overlap.

Vi Hart came up with this idea of snakes (see the link below).

Examples

			For n=2 the a(2)=1 there is only one snake:
(0,0), (0,1), (-1,1).
For n=3 the a(3)=2 there are two snakes:
(0,0), (0,1), (-1,1), (-1,0);
(0,0), (0,1), (-1,1), (-1,2).
Representing the walk (or snake) as a sequence of turns I and -I in the complex plane, with the initial condition that the first turn is I, for length 2 we have [I], for length 3 we have [I,I], [I,-I], and for length 4 we have [I,I,-I], [I,-I,I], [I,-I,-I].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 2..50
• Vi Hart, How To Snakes [Broken link?]
• Vi Hart, How to snakes, YouTube, March 2011.
• IBM Corp., Ponder This, April 2011.

Programs

Maple

ValidSnake:=proc(P) local S, visited, lastdir, lastpoint, j;
S:={0, 1}; lastdir:=1; lastpoint:=1;
for j from 1 to nops(P) do  lastdir:=lastdir*P[j];
  lastpoint:=lastpoint+lastdir;
  S:=S union {lastpoint};
od;
if (nops(S) = (2+nops(P))) then return(true); else return(false); fi;
end;
NextList:=proc(L) local S, snake, newsnake;
S:={ };
for snake in L do
  newsnake:=[op(snake), I];
  if ValidSnake(newsnake) then S:=S union {newsnake}; fi;
  newsnake:=[op(snake), -I];
  if ValidSnake(newsnake) then S:=S union {newsnake}; fi;
od;
return(S union { });
end;
L:={[I]}:
for k from 3 to 25 do
  L:=NextList(L):
  print(k, nops(L));
od:
# second Maple program:
a:= proc(n) local v, b;
      v:= proc() true end: v(0, 0), v(0, 1):= false$2:
      b:= proc(n, x, y, d) local c;
            if v(x, y) then v(x, y):= false;
               c:= `if`(n=0, 1,
                   `if`(d=1, b(n-1, x, y+1, 2) +b(n-1, x, y-1, 2),
                             b(n-1, x+1, y, 1) +b(n-1, x-1, y, 1) ));
               v(x, y):= true; c
            else 0 fi
          end;
      b(n-2, -1, 1, 1)
    end:
seq(a(n), n=2..25);  # Alois P. Heinz, Jun 10 2011

Mathematica

a[n_] := Module[{v, b}, v[, ] = True; v[0, 0] = v[0, 1] = False; b[m_, x_, y_, d_] := Module[{c}, If[v[x, y], v[x, y] = False; c = If[m == 0, 1, If[d == 1, b[m-1, x, y+1, 2] + b[m-1, x, y-1, 2], b[m-1, x+1, y, 1] + b[m-1, x-1, y, 1]]]; v[x, y] = True; c, 0]]; b[n-2, -1, 1, 1]]; Table[ a[n], {n, 2, 25}] (* Jean-François Alcover, Nov 07 2015, after Alois P. Heinz *)

Extensions

a(33)-a(40) from Alois P. Heinz, Jun 10 2011