%I #11 Jul 22 2022 09:50:24
%S 1,4,12,30,70,158,352,780,1724,3806,8398,18526,40864,90132,198796,
%T 438462,967062,2132926,4704320,10375708,22884348,50473022,111321758,
%U 245527870,541528768,1194379300,2634286476,5810101726,12814582758
%N Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.
%D J. Labelle, Paths in the Cartesian, triangular and hexagonal lattices, Bulletin of the ICA, 17, 1996, 47-61.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1,-1).
%F G.f.: (1+z^2)(1+z+z^2)/[(1-z)(1-2z-z^3)]= 1+2*(2+z^2)/((z-1)*(z^2+2*z-1)).
%F a(n) = 2*a(n-1) + a(n-3) + 6 for n >= 4.
%F a(n) = A008998(n+2) - A052980(n+1) - 3. - _Ralf Stephan_, May 15 2007
%F Conjecture: a(n) = A193641(n+2)-3, n>0 - _R. J. Mathar_, Jul 22 2022
%p g:=series((1+z^2)*(1+z+z^2)/(1-z)/(1-2*z-z^3),z=0,35): 1,seq(coeff(g,z^n), n=1..34);
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_, Dec 07 2004
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