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A284778 Number of self-avoiding planar walks of length n+1 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. 3
0, 1, 1, 4, 8, 22, 54, 142, 370, 983, 2627, 7086, 19238, 52561, 144377, 398518, 1104794, 3074809, 8588093, 24064642, 67630898, 190584766, 538412426, 1524554956, 4326119748, 12300296227, 35037658099, 99977847308, 285741659312, 817901027070, 2344475178110 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..2105

Alois P. Heinz, Animation of a(6)=54 walks

FORMULA

G.f.: (1-2*x-x^2-sqrt(1-4*x+2*x^2+4*x^3-3*x^4))/(2*(x+1)*x^3).

Recursion: see Maple program.

a(n) = A284414(n,n+1) = A284652(n,n+1).

a(n) ~ 3^(n+5/2) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 02 2017

MAPLE

a:= proc(n) option remember; `if`(n<3, (3-n)*n/2,

      ((n^2-n+3)*a(n-1)+(5*n-2)*n*a(n-2)+

       3*(n-1)*n*a(n-3))/((n+3)*(n-1)))

    end:

seq(a(n), n=0..35);

MATHEMATICA

CoefficientList[Series[(1 - 2*x - x^2 - Sqrt[1 - 4*x + 2*x^2 + 4*x^3 - 3*x^4])/(2*(x + 1)*x^3), {x, 0, 50}], x] (* Indranil Ghosh, Apr 02 2017 *)

CROSSREFS

First upper diagonal of A284414, A284652.

Sequence in context: A297339 A290138 A266922 * A057583 A129788 A170938

Adjacent sequences:  A284775 A284776 A284777 * A284779 A284780 A284781

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Apr 02 2017

STATUS

approved

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Last modified January 19 17:59 EST 2020. Contains 331051 sequences. (Running on oeis4.)