A271943 The sum of the widths of all bargraphs of semiperimeter n (n>=2).

1, 3, 10, 33, 108, 353, 1154, 3776, 12371, 40586, 133337, 438641, 1444848, 4764919, 15731660, 51993074, 172003177, 569531599, 1887392588, 6259572697, 20775058670, 68997611310, 229298384183, 762475061094, 2536834093693, 8444728118220, 28125035969635, 93713472090623, 312392935140250, 1041790050460247, 3475597146726072

Offset: 2

Author

Emeric Deutsch, May 21 2016

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
• A. Blecher, C. Brennan and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, 39 (2016), 619-635.
• A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A082582, A271942.

Programs

Maple

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);

Mathematica

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 *)

Formula

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)).

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