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A284230 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) with the restriction that (0,1) is never used below the diagonal and (1,0) is never used above the diagonal. 8
1, 2, 5, 24, 111, 762, 5127, 45588, 400593, 4370634, 47311677, 611446464, 7857786015, 117346361778, 1745000283087, 29562853594284, 499180661754849, 9458257569095826, 178734707493557301, 3744942786114870888, 78294815164675006479, 1797384789345147560298 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..448

Alois P. Heinz, Animation of a(4)=111 walks

Wikipedia, Lattice path

Wikipedia, Self-avoiding walk

FORMULA

a(n) ~ c * n^(n+2) / exp(n), where c = 0.7741273379869056907732932906458364317717498069987762339667734187318... - Vaclav Kotesovec, Mar 27 2017

Conjecture: a(n) -a(n-1) +(-n^2-n+3)*a(n-2) +(-n+2)*a(n-3) +(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Apr 09 2017

EXAMPLE

a(0) = 1: [(0,0)].

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

a(2) = 5: [(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)].

MAPLE

a:= proc(n) option remember; `if`(n<2, n+1,

      (n+irem(n, 2))*a(n-1)+(n-1)*a(n-2))

    end:

seq(a(n), n=0..25);

MATHEMATICA

a[n_]:=If[n<2, n + 1, (n + Mod[n, 2]) * a[n - 1] + (n - 1) a[n - 2]]; Table[a[n], {n, 0, 25}] (* Indranil Ghosh, Mar 27 2017 *)

CROSSREFS

Row sums of A284414.

Bisection (even part) gives A284461.

Cf. A001900, A277358, A284231, A285673.

Sequence in context: A012262 A012254 A322897 * A052111 A176473 A185056

Adjacent sequences:  A284227 A284228 A284229 * A284231 A284232 A284233

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Mar 23 2017

STATUS

approved

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Last modified July 4 04:47 EDT 2022. Contains 355064 sequences. (Running on oeis4.)