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A277359 Number of self-avoiding planar walks starting at (0,0), ending at (n,n), remaining in the first quadrant and using steps (0,1) and (1,0) on or below the diagonal and using steps (1,1), (-1,1), and (1,-1) on or above the diagonal. 6
1, 2, 7, 32, 176, 1126, 8227, 67768, 622706, 6323932, 70400734, 852952952, 11176241098, 157506733030, 2375966883371, 38200984291800, 652179787654530, 11783182484950980, 224623760504277810, 4505795627243046240, 94873821120923655336, 2092249161797280567516 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Both endpoints of each step have to satisfy the given restrictions.

a(n) is odd for n in {0, 2, 6, 14, 30, 62, ... } = { 2^n-2 | n>0 }.

LINKS

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

FORMULA

a(n) ~ exp(1)*(exp(1)-2) * n! * n. - Vaclav Kotesovec, Oct 13 2016

MAPLE

a:= proc(n) option remember; `if`(n<3, [1, 2, 7][n+1],

      ((n^3+10*n^2-10*n+1)*a(n-1)-(2*(4*n^3+2*n^2-29*n+28))

        *a(n-2)+(4*(n-2))*(2*n-3)^2*a(n-3))/(n*(n+1)))

    end:

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

MATHEMATICA

a[n_] := a[n] = If[n<3, {1, 2, 7}[[n+1]], ((n^3+10*n^2-10*n+1)*a[n-1] - (2*(4*n^3+2*n^2-29*n+28))*a[n-2] + (4*(n-2))*(2*n-3)^2*a[n-3])/(n*(n+1)) ]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Jan 25 2017, translated from Maple *)

CROSSREFS

Cf. A000108, A000142, A000918, A277175, A277176, A277358, A277360, A277756.

Sequence in context: A161393 A143426 A125223 * A005362 A059439 A190123

Adjacent sequences:  A277356 A277357 A277358 * A277360 A277361 A277362

KEYWORD

nonn,walk

AUTHOR

Alois P. Heinz, Oct 10 2016

STATUS

approved

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Last modified March 20 09:50 EDT 2019. Contains 321345 sequences. (Running on oeis4.)