A100693 Number of self-avoiding paths with n steps on a hexagonal lattice in the strip Z x {0,1,2}.

1, 2, 3, 5, 6, 7, 9, 14, 14, 14, 22, 30, 28, 28, 44, 60, 56, 56, 88, 120, 112, 112, 176, 240, 224, 224, 352, 480, 448, 448, 704, 960, 896, 896, 1408, 1920, 1792, 1792, 2816, 3840, 3584, 3584, 5632, 7680, 7168, 7168, 11264, 15360, 14336, 14336, 22528, 30720

Offset

0,2

References

J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.

Links

Table of n, a(n) for n=0..51.

Formula

G.f.: (1+2z+3z^2+5z^3+4z^4+3z^5+3z^6+4z^7+2z^8+4z^10+2z^11)/(1-2z^4).

For n>=2: a(4n) = a(4n+1) = 7*2^(n-1), a(4n+2) = 11*2^(n-1), a(4n+3) = 15*2^(n-1).

Maple

g:=series((1+2*z+3*z^2+5*z^3+4*z^4+3*z^5+3*z^6+4*z^7+2*z^8+4*z^10+2*z^11)/(1-2*z^4), z=0, 64): 1, seq(coeff(g, z^n), n=1..60);

Mathematica

a[n_] := If[n <= 7, {1, 2, 3, 5, 6, 7, 9, 14}[[n+1]], Switch[Mod[n, 4], 0, 7*2^(n/4-1), 1, 7*2^((n-5)/4), 2, 11*2^((n-6)/4), 3, 15*2^((n-7)/4)]]; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Jul 09 2017 *)

Crossrefs

Sequence in context: A056900 A226808 A096594 * A030159 A030161 A361372

Adjacent sequences: A100690 A100691 A100692 * A100694 A100695 A100696

Keyword

nonn

Author

Emeric Deutsch, Dec 07 2004

Status

approved