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

Author

N. J. A. Sloane

Keywords

• nonn
• walk

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).

Crossrefs

Cf. A001411, A002900, A078797.

Programs

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 *)

Formula

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