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%I #14 Sep 08 2022 08:45:15
%S 1,3,4,4,6,10,10,8,12,20,20,16,24,40,40,32,48,80,80,64,96,160,160,128,
%T 192,320,320,256,384,640,640,512,768,1280,1280,1024,1536,2560,2560,
%U 2048,3072,5120,5120,4096,6144,10240,10240,8192,12288,20480,20480,16384
%N Number of self-avoiding paths with n steps on a hexagonal lattice in the strip Z X {-1,0,1}.
%D J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.
%H G. C. Greubel, <a href="/A100692/b100692.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,2).
%F G.f.: (1+3*z+4*z^2+4*z^3+4*z^4+4*z^5+2*z^6) / (1-2*z^4).
%F 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
%p 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);
%t 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 *)
%o (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
%o (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
%o (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
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Dec 07 2004
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