A046661 Number of n-step self-avoiding walks on the square lattice with first step specified.

1, 3, 9, 25, 71, 195, 543, 1479, 4067, 11025, 30073, 81233, 220375, 593611, 1604149, 4311333, 11616669, 31164683, 83779155, 224424291, 602201507, 1611140121, 4316653453, 11536599329, 30870338727, 82428196555, 220329372907

Offset

1,2

Comments

Used as the denominator for the mean square displacement of all different self-avoiding n-step walks in A078797. - Hugo Pfoertner, Dec 09 2002

Number of ways a toy snake with n segments can be bent without flipping the snake upside down. Each segment must be perpendicular or parallel with each adjacent segment. A "slither" is a way of writing down the configuration of a snake; starting from the tail, write down which direction the next segment is pointing (R for right, S for straight, L for left). E.g., a snake with 10 segments may have the valid slither RLRRLLRRL, but not RSRRSSLSL.

Links

Hugo Pfoertner, Table of n, a(n) for n = 1..79

G. T. Barkema and S. Flesia, Two-dimensional oriented self-avoiding walks with parallel contacts, J. Stat. Phys. 85 (1996) no 3/4, 363-381. [a(30) to a(34)]

D. Bennet-Wood, J. L. Cardy, S. Flesia, A. J. Guttmann et al., Oriented self-avoiding walks with orientation-dependent interactions, J. Phys. A: Math. Gen. 28 (1995) no 18, 5143-5163. [up to a(29)]

V. Hart, How to Snakes, Youtube Video (2011).

Formula

a(n) = A001411(n)/4 = A002900(n)/2.

Mathematica

(* b = A001411 *) mo = Tuples[{-1, 1}, 2]; b[0] = 1; b[tg_, p_:{{0, 0}}] := b[tg, p] = Block[{e, mv = Complement[Last[p] + #& /@ mo, p]}, If[tg == 1, Length[mv], Sum[b[tg - 1, Append[p, e]], {e, mv}]]];
a[n_] := b[n]/4;
Table[an = a[n]; Print[an]; an, {n, 1, 16}] (* Jean-François Alcover, Nov 02 2018, after Giovanni Resta in A001411 *)

Crossrefs

Cf. A001411, A002900, A078797.

Sequence in context: A211283 A333608 A058719 * A309105 A101197 A233828

Adjacent sequences: A046658 A046659 A046660 * A046662 A046663 A046664

Keyword

nonn,walk

Author

N. J. A. Sloane

Status

approved