A284461 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

1, 5, 111, 5127, 400593, 47311677, 7857786015, 1745000283087, 499180661754849, 178734707493557301, 78294815164675006479, 41186656484051421462615, 25619826402721039367943729, 18600984174200732870460447213, 15588291843672510150758754601407

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..224

Alois P. Heinz, Animation of a(2)=111 walks

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

Formula

a(n) = A284230(2n).

a(n) = Sum_{k=2n..n*(2n+3)} A284414(2n,k).

Maple

b:= proc(n) option remember; `if`(n<2, n+1,
      (n+irem(n, 2))*b(n-1)+(n-1)*b(n-2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n+1,
      ((2*n+1)^2-2)*a(n-1)-(4*n-6)*n*a(n-2))
    end:
seq(a(n), n=0..15);

Mathematica

a[n_] := a[n] = If[n<2, 4n+1, ((2n+1)^2-2) a[n-1] - (4n-6) n a[n-2]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jun 19 2017, after 2nd Maple program *)

Crossrefs

Bisection of A284230 (even part).

Cf. A284414, A285673.

Sequence in context: A294965 A219161 A367248 * A002400 A268404 A258795

Adjacent sequences: A284458 A284459 A284460 * A284462 A284463 A284464

Keyword

nonn,walk

Author

Alois P. Heinz, Mar 27 2017

Status

approved