A317985 -id:A317985 - cp's OEIS frontend

A277358 Number of 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, 2, 9, 58, 491, 5142, 64159, 929078, 15314361, 283091122, 5799651689, 130423248378, 3193954129651, 84607886351462, 2410542221526399, 73500777054712438, 2388182999073694001, 82374234401380995042, 3006071549433968619529, 115713455097715665452858

Offset: 0

Alois P. Heinz, Oct 10 2016

Alois P. Heinz, Table of n, a(n) for n = 0..403

Cf. A277359, A277360, A277424, A284230, A317985.

a:= n-> n!*coeff(series(exp(-x/2)/(1-2*x)^(5/4), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n+1,
       2*n*a(n-1) +(n-1)*a(n-2))
    end:
seq(a(n), n=0..25);

a[n_] := a[n] = If[n < 2, n+1, 2*n*a[n-1] + (n-1)*a[n-2]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 29 2017, translated from Maple *)

E.g.f.: exp(-x/2)/(1-2*x)^(5/4).
a(n) = 2*n*a(n-1) + (n-1)*a(n-2) for n>1, a(0)=1, a(1)=2.
a(n) ~ sqrt(Pi) * 2^(n+5/2) * n^(n+3/4) / (Gamma(1/4) * exp(n+1/4)). - Vaclav Kotesovec, Oct 13 2016

A320512 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) such that (0,1) is never used directly before or after (1,0) or (1,1).

Original entry on oeis.org

1, 5, 31, 258, 2702, 33821, 492978, 8198218, 153136209, 3173544162, 72241986729, 1791612993205, 48074653669593, 1387590910289915, 42863756641047136, 1410904918289665343, 49296029555617568097, 1822020250023113834772, 71023629427964322798782

Offset: 0

Alois P. Heinz, Oct 22 2018

Alois P. Heinz, Table of n, a(n) for n = 0..402

Cf. A317985.

b:= proc(x, y, i) option remember; (l-> `if`(min(x, y)<0, 0,
      `if`(max(x, y)=0, [1$2], add(`if`({i, j} in {{1, 2}, {3, 5},
       {4, 5}}, 0, (p-> p+[0, p[1]])(b(x-l[j][1], y-l[j][2], j))),
       j=1..5))))([[-1, 1], [1, -1], [1, 1], [1, 0], [0, 1]])
    end:
a:= n-> b(n, 0$2)[2]:
seq(a(n), n=0..20);

b[x_, y_, i_] := b[x, y, i] = With[{l = {{-1, 1}, {1, -1}, {1, 1}, {1, 0}, {0, 1}}}, If[Min[x, y] < 0, {0, 0}, If[Max[x, y] == 0, {1, 1}, Sum[If[ MemberQ[{{1, 2}, {3, 5}, {4, 5}}, Sort@{i, j}], {0, 0}, Function[p, p + {0, p[[1]]}][b[x - l[[j]][[1]], y - l[[j]][[2]], j]]], {j, 5}]]]];
a[n_] := b[n, 0, 0][[2]];
a /@ Range[0, 20] (* Jean-François Alcover, May 14 2020, after Maple *)

a(n) ~ c * n! * 2^n * n^(7/4), where c = 0.1758027947... - Vaclav Kotesovec, May 14 2020

cp's OEIS Frontend

A277358 Number of 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).

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