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%I #40 Mar 22 2020 00:44:55
%S 1,3,9,25,71,195,543,1479,4067,11025,30073,81233,220375,593611,
%T 1604149,4311333,11616669,31164683,83779155,224424291,602201507,
%U 1611140121,4316653453,11536599329,30870338727,82428196555,220329372907
%N Number of n-step self-avoiding walks on the square lattice with first step specified.
%C Used as the denominator for the mean square displacement of all different self-avoiding n-step walks in A078797. - _Hugo Pfoertner_, Dec 09 2002
%C 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.
%H Hugo Pfoertner, <a href="/A046661/b046661.txt">Table of n, a(n) for n = 1..79</a>
%H G. T. Barkema and S. Flesia, <a href="http://dx.doi.org/10.1007/BF02174210">Two-dimensional oriented self-avoiding walks with parallel contacts</a>, J. Stat. Phys. 85 (1996) no 3/4, 363-381. [a(30) to a(34)]
%H D. Bennet-Wood, J. L. Cardy, S. Flesia, A. J. Guttmann et al., <a href="http://dx.doi.org/10.1088/0305-4470/28/18/007">Oriented self-avoiding walks with orientation-dependent interactions</a>, J. Phys. A: Math. Gen. 28 (1995) no 18, 5143-5163. [up to a(29)]
%H V. Hart, <a href="http://youtu.be/Gx5D09s5X6U">How to Snakes</a>, Youtube Video (2011).
%F a(n) = A001411(n)/4 = A002900(n)/2.
%t (* 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}]]];
%t a[n_] := b[n]/4;
%t Table[an = a[n]; Print[an]; an, {n, 1, 16}] (* _Jean-François Alcover_, Nov 02 2018, after _Giovanni Resta_ in A001411 *)
%Y Cf. A001411, A002900, A078797.
%K nonn,walk
%O 1,2
%A _N. J. A. Sloane_
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