A100692 Number of self-avoiding paths with n steps on a hexagonal lattice in the strip Z X {-1,0,1}.

1, 3, 4, 4, 6, 10, 10, 8, 12, 20, 20, 16, 24, 40, 40, 32, 48, 80, 80, 64, 96, 160, 160, 128, 192, 320, 320, 256, 384, 640, 640, 512, 768, 1280, 1280, 1024, 1536, 2560, 2560, 2048, 3072, 5120, 5120, 4096, 6144, 10240, 10240, 8192, 12288, 20480, 20480, 16384

Offset

0,2

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,2).

Formula

G.f.: (1+3*z+4*z^2+4*z^3+4*z^4+4*z^5+2*z^6) / (1-2*z^4).

a(0)=1, a(1)=3, a(2)=4, a(4*n+3)=4*2^n, a(4*n+4)=6*2^n, a(4*n+5)=a(4*n+6)=10*2^n. - Ralf Stephan, May 16 2007

Maple

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

Mathematica

CoefficientList[Series[(1 +3*x +4*x^2 +4*x^3 +4*x^4 +4*x^5 +2*x^6)/(1 - 2*x^4), {x, 0, 60}], x] (* G. C. Greubel, May 21 2019 *)

Prog

(PARI) my(x='x+O('x^60)); Vec( (1 +3*x +4*x^2 +4*x^3 +4*x^4 +4*x^5 +2*x^6)/(1-2*x^4) ) \\ G. C. Greubel, May 21 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( (1 +3*x +4*x^2 +4*x^3 +4*x^4 +4*x^5 +2*x^6)/(1-2*x^4) )); // G. C. Greubel, May 21 2019
(Sage) ((1 +3*x +4*x^2 +4*x^3 +4*x^4 +4*x^5 +2*x^6)/(1-2*x^4)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 21 2019

Crossrefs

Sequence in context: A047877 A351371 A280448 * A360724 A089640 A086659

Adjacent sequences: A100689 A100690 A100691 * A100693 A100694 A100695

Keyword

nonn,easy

Author

Emeric Deutsch, Dec 07 2004

Status

approved