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%I #35 Jul 26 2022 13:03:31
%S 1,3,10,33,108,353,1154,3776,12371,40586,133337,438641,1444848,
%T 4764919,15731660,51993074,172003177,569531599,1887392588,6259572697,
%U 20775058670,68997611310,229298384183,762475061094,2536834093693,8444728118220,28125035969635,93713472090623,312392935140250,1041790050460247,3475597146726072
%N The sum of the widths of all bargraphs of semiperimeter n (n>=2).
%C The number of level steps in all bargraphs of semiperimeter n+1 for n>=2. A level step is a pair of adjacent horizontal steps. - _Arnold Knopfmacher_, Nov 04 2016
%H A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, <a href="http://dx.doi.org/10.1016/j.dam.2014.08.026">The height and width of bargraphs</a>, Discrete Applied Math. 180, (2015), 36-44.
%H A. Blecher, C. Brennan and A. Knopfmacher, <a href="http://www.tandfonline.com/doi/abs/10.2989/16073606.2015.1121932">Combinatorial parameters in bargraphs</a>, Quaestiones Mathematicae, 39 (2016), 619-635.
%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="http://dx.doi.org/10.1080/0035919X.2015.1059905">Peaks in bargraphs</a>, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
%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 G.f.: (1 - z)*(1 - 2*z - z^2 - sqrt(1 - 4*z + 2*z^2 + z^4))/(2*z*sqrt(1 - 4*z + 2*z^2 + z^4)).
%F a(n) = Sum_{k>=1} k*A271942(n,k). D-finite with recurrence (n+1)*a(n) +(-6*n+1)*a(n-1) +(9*n-14)*a(n-2) -2*a(n-3) +(-n+11)*a(n-4) +(-2*n+9)*a(n-5) +(-n+6)*a(n-6)=0. - _R. J. Mathar_, Jun 02 2016
%e a(4)=10 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, the sum of their widths is 3+2+2+2+1=10.
%p g := (1/2)*(1-z)*(1-2*z-z^2-sqrt(1-4*z+2*z^2+z^4))/(z*sqrt(1-4*z+2*z^2+z^4)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 2 .. 35);
%t Drop[CoefficientList[Series[(1 - x) (1 - 2 x - x^2 - Sqrt[1 - 4 x + 2 x^2 + x^4])/(2 x Sqrt[1 - 4 x + 2 x^2 + x^4]), {x, 0, 32}], x], 2] (* _Michael De Vlieger_, May 21 2016 *)
%Y Cf. A082582, A271942.
%K nonn
%O 2,2
%A _Emeric Deutsch_, May 21 2016