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%I #28 Nov 27 2017 05:49:45
%S 2,6,16,39,92,211,476,1059,2332,5091,11036,23779,50972,108771,231196,
%T 489699,1034012,2177251,4572956,9582819,20039452,41826531,87148316,
%U 181287139,376555292,781072611,1618069276,3347986659,6919669532,14286731491,29468247836,60726065379,125031270172,257220819171,528758195996
%N The sum of the semiperimeters of the bargraphs of area n (n>=1).
%H A. Blecher, C. Brennan, and A. Knopfmacher, <a href="https://doi.org/10.2989/16073606.2015.1121932">Combinatorial parameters in bargraphs</a>, Quaestiones Mathematicae, Volume 39, 2016 - Issue 5.
%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.
%H M. Bousquet-Mélou and R. Brak, <a href="https://hal.archives-ouvertes.fr/hal-00342024">Exactly solved models of polyominoes and polygons</a>, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.
%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-4,4).
%F G.f.: g = t(2-2t-2t^2+t^3)/((1-t^2)(1-2t)^2).
%F a(n) = (15*n2^n+29*2^n-2(-1)^n-18)/36.
%F a(n) = Sum_{k>=2} k * A273346(k,n).
%e a(4) = 39 because the 8 bargraphs of area 4 correspond to the compositions [2,2],[4],[3,1],[1,3],[2,1,1],[1,2,1],[1,1,2],[1,1,1,1] and the sum of their semiperimeters is 4 + 7*5 = 39.
%p a := proc(n) (5/12)*n*2^n+(29/36)*2^n-(1/18)*(-1)^n-1/2 end proc:
%p seq(a(n), n = 1 .. 35);
%t LinearRecurrence[{4, -3, -4, 4}, {2, 6, 16, 39}, 35] (* _Jean-François Alcover_, Nov 27 2017 *)
%o (PARI) first(n) = Vec(x*(2-2*x-2*x^2+x^3)/((1-x^2)*(1-2*x)^2) + O(x^(n+1))) \\ _Iain Fox_, Nov 27 2017
%Y Cf. A273346, A273347.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Jun 03 2016
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