A273348 The sum of the semiperimeters of the bargraphs of area n (n>=1).
2, 6, 16, 39, 92, 211, 476, 1059, 2332, 5091, 11036, 23779, 50972, 108771, 231196, 489699, 1034012, 2177251, 4572956, 9582819, 20039452, 41826531, 87148316, 181287139, 376555292, 781072611, 1618069276, 3347986659, 6919669532, 14286731491, 29468247836, 60726065379, 125031270172, 257220819171, 528758195996
Offset: 1
Keywords
Examples
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.
Links
- A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, Volume 39, 2016 - Issue 5.
- M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
- M. Bousquet-Mélou and R. Brak, Exactly solved models of polyominoes and polygons, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.
- Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
- Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).
Programs
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Maple
a := proc(n) (5/12)*n*2^n+(29/36)*2^n-(1/18)*(-1)^n-1/2 end proc: seq(a(n), n = 1 .. 35);
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Mathematica
LinearRecurrence[{4, -3, -4, 4}, {2, 6, 16, 39}, 35] (* Jean-François Alcover, Nov 27 2017 *) -
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
Formula
G.f.: g = t(2-2t-2t^2+t^3)/((1-t^2)(1-2t)^2).
a(n) = (15*n2^n+29*2^n-2(-1)^n-18)/36.
a(n) = Sum_{k>=2} k * A273346(k,n).