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A006189 Number of self-avoiding walks of any length from NW to SW corners of a grid or lattice with n rows and 3 columns.
(Formerly M2891)
8
1, 1, 3, 11, 38, 126, 415, 1369, 4521, 14933, 49322, 162900, 538021, 1776961, 5868903, 19383671, 64019918, 211443426, 698350195, 2306494009, 7617832221, 25159990673, 83097804242, 274453403400, 906458014441, 2993827446721, 9887940354603, 32657648510531 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) = number of non-self-intersecting (or self-avoiding) paths from upper-left to lower-left of a grid of squares with 3 columns and n rows. E.g., for 3 columns and 2 rows, the paths are D, RDL, and RRDLL and the second a(n) = 3. The next a(n) = 11, which is the number of paths in a 3 X 3 grid: DD, DRDL, DRRDLL, DRURDDLL, RDDL, RDRDLL, RDLD, RRDDLL, RRDDLULD, RRDLDL, RRDLLD (where R=right, L=left, D=down, U=up). - Toby Gottfried, Mar 04 2013

REFERENCES

H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..100

H. L. Abbott and D. Hanson, A lattice path problem, Ars Combin., 6 (1978), 163-178. (Annotated scanned copy)

Index entries for linear recurrences with constant coefficients, signature (4,-3,2,1).

FORMULA

a(n) = 4*a(n-1) - 3*a(n-2) + 2*a(n-3) + a(n-4) for n > 3. - Giovanni Resta, Mar 13 2013

G.f.: (1 - x)*(1 - 2*x) / ((1 - x + x^2)*(1 - 3*x - x^2)). - Colin Barker, Nov 17 2017

2*a(n) = A010892(n) +A052924(n). - R. J. Mathar, Sep 27 2020

MATHEMATICA

(* Checked against 100 terms of b-file *) LinearRecurrence[{4, -3, 2, 1}, {1, 3, 11, 38}, 100] (* Jean-Fran├žois Alcover, Oct 08 2017 *)

PROG

(PARI) Vec((1 - x)*(1 - 2*x) / ((1 - x + x^2)*(1 - 3*x - x^2)) + O(x^40)) \\ Colin Barker, Nov 17 2017

CROSSREFS

Column 3 of A271465.

Cf. A005409 (grids with 3 rows), A001333.

Cf. A214931 (grids with 4 rows).

Cf. A216211 (grids with 4 columns).

Sequence in context: A265796 A129962 A026361 * A092201 A273526 A026943

Adjacent sequences:  A006186 A006187 A006188 * A006190 A006191 A006192

KEYWORD

nonn,walk,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Based on upper-left to lower-left path-counting program, more terms from Toby Gottfried, Mar 04 2013

Name clarified, offset changed, a(16)-a(25) from Andrew Howroyd, Apr 07 2016

a(0)=1 prepended by Colin Barker, Nov 17 2017

STATUS

approved

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Last modified November 28 02:22 EST 2020. Contains 338699 sequences. (Running on oeis4.)