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%I #11 Jul 22 2022 10:20:49
%S 0,1,2,3,6,13,27,57,123,267,584,1289,2864,6399,14373,32435,73498,
%T 167175,381551,873541,2005622,4616895,10653607,24638263,57097885,
%U 132575577,308378460,718506295,1676706422,3918515001,9170350093,21488961641,50417138776,118425429213,278476687643
%N Number of bargraphs of semiperimeter n having no horizontal segments of length 1 (n>=2). By a horizontal segment of length 1 we mean a horizontal step that is not adjacent to any other horizontal step.
%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%F a(n) = A274491(n,0).
%F G.f.: g(z)=(1-2z+z^2-2z^3-sqrt((1-z)(1-3z+3z^2-5z^3+4z^4-4z^5)))/(2z^2).
%F D-finite with recurrence (n+2)*a(n) +2*(-2*n-1)*a(n-1) +6*(n-1)*a(n-2) +4*(-2*n+5)*a(n-3) +9*(n-4)*a(n-4) +4*(-2*n+11)*a(n-5) +4*(n-7)*a(n-6)=0. - _R. J. Mathar_, Jul 22 2022
%e a(4)=2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding pictures give the values 0,2,2,0,1 for the number of horizontal segments of length 1.
%p g:=((1-2*z+z^2-2*z^3-sqrt((1-z)*(1-3*z+3*z^2-5*z^3+4*z^4-4*z^5)))*(1/2))/z^2: gser:=series(g,z=0,40): seq(coeff(gser,z,n),n=2..36);
%Y Cf. A082582, A274491.
%K nonn,easy
%O 2,3
%A _Emeric Deutsch_ and _Sergi Elizalde_, Jun 27 2016
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