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 A192871 Number of n-step prudent self-avoiding walks on honeycomb lattice. 2
 1, 3, 6, 12, 24, 48, 90, 168, 318, 594, 1092, 2004, 3678, 6720, 12210, 22128, 40074, 72372, 130380, 234432, 421128, 755208, 1352328, 2418246, 4320552, 7709898, 13744764, 24477618, 43560444, 77448330, 137602440, 244277016, 433399824, 768379830, 1361530134 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A prudent walk never takes a step pointing towards a vertex it has already visited.  Prudent walks are self-avoiding but not reversible in general. REFERENCES M. Bousquet-Mélou, Families of prudent self-avoiding walks, DMTCS proc. AJ, 2008, 167-180. E. Duchi, On some classes of prudent walks, in: FPSAC'05, Taormina, Italy, 2005. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..110 EXAMPLE This 8-step prudent self-avoiding walk on honeycomb lattice from (S) to (E) is not reversible: .           o...o       o...o .          .     .     .     . .     o...o       4---3       o .    .     .     /     \     . .   o       6---5       2...o .    .     /     .     /     . .     o...7      (S)--1       o .    .     \     .     .     . .   o      (E)..o       o...o .    .     .     .     . .     o...o       o...0 MAPLE i:= n-> max(n, 0)+1: d:= n-> max(n-1, -1): b:= proc(n, x, y, z, u, v, w) option remember;     `if`(n=0, 1, `if`(x>y, b(n, y, x, w, v, u, z),     `if`(min(y, z)<=0 or x=-1,         b(n-1, d(y), d(z), u, i(v), i(w), x), 0)+     `if`(min(w, x)<=0 or y=-1,         b(n-1, d(w), d(x), y, i(z), i(u), v), 0)))     end: a:= n-> `if`(n<2, 1 +2*n, 6*b(n-2, -1, -1, 1, 2, 1, -1)): seq(a(n), n=0..20); MATHEMATICA i[n_] := Max[n, 0] + 1; d[n_] := Max[n - 1, -1]; b[n_, x_, y_, z_, u_, v_, w_] := b[n, x, y, z, u, v, w] = If[n == 0, 1, If[x > y, b[n, y, x, w, v, u, z], If[Min[y, z] <= 0 || x == -1, b[n - 1, d[y], d[z], u, i[v], i[w], x], 0] + If[Min[w, x] <= 0 || y == -1, b[n - 1, d[w], d[x], y, i[z], i[u], v], 0]]]; a[n_] := If[n < 2, 1 + 2 n, 6 b[n - 2, -1, -1, 1, 2, 1, -1]]; a /@ Range[0, 34] (* Jean-François Alcover, Sep 22 2019, after Alois P. Heinz *) CROSSREFS Cf. A001668, A006851, A192208. Sequence in context: A115807 A322676 A102255 * A002910 A001668 A080616 Adjacent sequences:  A192868 A192869 A192870 * A192872 A192873 A192874 KEYWORD nonn,walk AUTHOR Alois P. Heinz, Jul 11 2011 STATUS approved

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Last modified June 24 02:56 EDT 2021. Contains 345415 sequences. (Running on oeis4.)