A277424 Total number of nodes summed over all self-avoiding planar walks starting at (0,0), ending at (n,0), remaining in the first quadrant and using steps (0,1), (1,0), (1,1), (-1,1), and (1,-1).

1, 5, 39, 379, 4457, 61503, 974107, 17412317, 346662981, 7605810685, 182298744203, 4738700778123, 132767583248917, 3988244997744743, 127859570155253607, 4357113615504651565, 157266354405499307369, 5993377455733610208885, 240479249123008267155343

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

From Vaclav Kotesovec, Oct 14 2016: (Start)

Recurrence: (n^3 + n^2 - 6*n + 1)*a(n) = (4*n^4 + 6*n^3 - 28*n^2 + 3*n + 4)*a(n-1) - (4*n^5 + 2*n^4 - 42*n^3 + 50*n^2 - 13)*a(n-2) - (n-2)*(4*n^4 + 8*n^3 - 34*n^2 - n + 16)*a(n-3) - (n-3)*(n-2)*(n^3 + 4*n^2 - n - 3)*a(n-4).

a(n) ~ sqrt(Pi) * 2^(n+3/2) * n^(n+11/4) / (3 * Gamma(1/4) * exp(n+1/4)).

(End)

Maple

b:= proc(x, y, t) option remember; `if`(x<0 or y<0, 0,
      `if`(x=0 and y=0, [1$2], (p-> p+ [0, p[1]])(
       b(x-1, y, 0)+ b(x, y-1, 0)+ b(x-1, y-1, 0)+
      `if`(t<>2, b(x+1, y-1, 1), 0)+
      `if`(t<>1, b(x-1, y+1, 2), 0))))
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..25);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[x < 0 || y < 0, 0, If[x == 0 && y == 0, {1, 1}, Function[p, p + {0, p[[1]]}][b[x - 1, y, 0] + b[x, y - 1, 0] + b[x - 1, y - 1, 0] + If[t != 2, b[x + 1, y - 1, 1], 0] + If[t != 1, b[x - 1, y + 1, 2], 0]]]];
a[n_] := b[n, 0, 0][[2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 19 2017, translated from Maple *)

Crossrefs

Cf. A277358, A284231, A285673.

Sequence in context: A365765 A247772 A129763 * A182954 A215506 A316654

Adjacent sequences: A277421 A277422 A277423 * A277425 A277426 A277427

Keyword

nonn

Author

Alois P. Heinz, Oct 14 2016

Status

approved