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A216211 Number of self-avoiding walks of any length from NW to SW corners of a grid or lattice with n rows and 4 columns. 5

%I

%S 1,4,28,178,1008,5493,29879,163357,895519,4911542,26932856,147666219,

%T 809584243,4438588016,24334993398,133419407518,731487440774,

%U 4010463570150,21987820817522,120550714106036,660932932241338,3623639639745022,19867014703421770,108923158026586497,597183548915194615

%N Number of self-avoiding walks of any length from NW to SW corners of a grid or lattice with n rows and 4 columns.

%C As n increases, the ratio of a(n)/a(n-1) appears to converge to around 5.483.

%H Andrew Howroyd, <a href="/A216211/b216211.txt">Table of n, a(n) for n = 1..100</a>

%F Conjectures from _Colin Barker_, Nov 18 2017: (Start)

%F G.f.: x*(1 - 8*x + 34*x^2 - 66*x^3 + 21*x^4 + 85*x^5 - 64*x^6 - 45*x^7 + 26*x^8 + 11*x^9 - 3*x^10 - x^11) / ((1 - 8*x + 15*x^2 - 5*x^3 - 9*x^4 + 2*x^5 + x^6)*(1 - 4*x + 7*x^2 - 3*x^3 - 7*x^4 + 2*x^5 + x^6)).

%F a(n) = 12*a(n-1) - 54*a(n-2) + 124*a(n-3) - 133*a(n-4) - 16*a(n-5) + 175*a(n-6) - 94*a(n-7) - 69*a(n-8) + 40*a(n-9) + 12*a(n-10) - 4*a(n-11) - a(n-12) for n>12.

%F (End)

%e For n=2, using the notation D(own), R(ight), L(eft), U(p), the 4 walks are {D, RDL, RRDLL, RRRDLLL}.

%t a[n_] := Block[{t=0,w,b=Array[1&, {n,4}]}, w[rr_,cc_] := Block[{r,c}, If[rr+cc == 2, t++, Do[{r,c} = {rr,cc} + e; If[0<c<5 && 0<r<=n && b[[r,c]] > 0, b[[r,c]] = 0; w[r, c]; b[[r,c]] = 1], {e, {{-1,0}, {1,0}, {0,1}, {0,-1}}}]]]; b[[n,1]] = 0; w[n,1]; t]; a /@ Range[6] (* _Giovanni Resta_, Mar 13 2013 *)

%Y Column 4 of A271465. Cf. A005409 for grids with 3 rows, A006189 for grids with 3 columns, and A214931 for grids with 4 rows.

%K nonn,walk

%O 1,2

%A _Toby Gottfried_, Mar 13 2013

%E a(13)-a(14) from _Giovanni Resta_, Mar 13 2013

%E Terms a(15) and beyond from _Andrew Howroyd_, Apr 08 2016

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Last modified August 20 03:48 EDT 2019. Contains 326139 sequences. (Running on oeis4.)