A277424 -id:A277424 - 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

A284231 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) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal.

Original entry on oeis.org

1, 5, 21, 152, 975, 8835, 75499, 830180, 8819417, 114384573, 1450018173, 21689509992, 319180726887, 5411092531323, 90615453774771, 1717272516535812, 32234085990345105, 675335923050095253, 14040521125141683717, 322252846702242056280, 7349647183279936080543

Offset: 0

Alois P. Heinz, Mar 23 2017

			a(0) = 1: [(0,0)].
a(1) = 5: [(0,0),(1,0)], [(0,0),(0,1),(1,0)].
a(2) = 21: [(0,0),(1,0),(2,0)], [(0,0),(0,1),(1,0),(2,0)], [(0,0),(1,1),(2,0)], [(0,0),(0,1),(0,2),(1,1),(2,0)], [(0,0),(1,0),(0,1),(0,2),(1,1),(2,0)].

Alois P. Heinz, Table of n, a(n) for n = 0..446
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A277424, A284230, A284414, A285673.

a(n) = Sum_{k=n..n*(n+3)/2} (k+1) * A284414(n,k).

A285673 Total number of nodes summed over all 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.

Original entry on oeis.org

1, 20, 907, 69928, 8190329, 1352590668, 299134112595, 85301875065360, 30466886170947633, 13319092946564641476, 6994728861780241970523, 4344874074153003071077560, 3150737511338249699332032297, 2637670112785000275509973725820, 2524664376417193478764383143006883

Offset: 0

Alois P. Heinz, Apr 24 2017

Alois P. Heinz, Table of n, a(n) for n = 0..223
Wikipedia, Lattice path
Wikipedia, Self-avoiding walk

Cf. A277424, A277756, A284230, A284231, A284461.

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]])(
      `if`(x>y,  b(x-1, y,   0), 0)+
      `if`(y>x,  b(x,   y-1, 0), 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$2, 0)[2]:
seq(a(n), n=0..20);

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]]}][If[x > y,  b[x - 1, y,   0], 0] + If[y > x,  b[x, y - 1, 0], 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, n, 0][[2]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 19 2017, translated from Maple *)

Recurrence: (768*n^7 - 9760*n^6 + 42960*n^5 - 72624*n^4 + 4272*n^3 + 120634*n^2 - 117042*n + 29523)*a(n) = 4*(1536*n^9 - 17216*n^8 + 56928*n^7 - 19536*n^6 - 199576*n^5 + 257144*n^4 + 67826*n^3 - 200220*n^2 + 46970*n - 201)*a(n-1) - (12288*n^11 - 143872*n^10 + 517376*n^9 - 304896*n^8 - 1803648*n^7 + 3174144*n^6 - 434416*n^5 - 1420224*n^4 - 672608*n^3 + 1216378*n^2 - 69926*n - 51561)*a(n-2) + 8*(n-1)*(3072*n^10 - 40576*n^9 + 179200*n^8 - 212640*n^7 - 583984*n^6 + 1881504*n^5 - 1496616*n^4 - 314158*n^3 + 703776*n^2 - 93829*n - 15912)*a(n-3) - 4*(n-2)*(n-1)*(2*n - 9)*(2*n - 7)*(768*n^7 - 4384*n^6 + 528*n^5 + 22656*n^4 - 24944*n^3 - 2966*n^2 + 8162*n - 1269)*a(n-4). - Vaclav Kotesovec, Apr 25 2017

a(n) ~ c * n^(2*n+4) * 2^(2*n) / exp(2*n), where c = 2.064339567965... - Vaclav Kotesovec, Apr 25 2017

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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