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A271943 The sum of the widths of all bargraphs of semiperimeter n (n>=2). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

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

LINKS

Table of n, a(n) for n=2..32.

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.

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).

Conjecture: (n+1)*(4*n^2-16*n+11)*a(n) +(-16*n^3+60*n^2-32*n+11) *a(n-1) +(8*n^3-44*n^2+42*n+17) *a(n-2) +(-4*n^2+12*n+11)*a(n-3) +(n-4) *(4*n^2-8*n-1) *a(n-4)=0. - R. J. Mathar, Jun 02 2016

Conjecture: (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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A082582, A271942.

Sequence in context: A292397 A060557 A018920 * A255116 A006190 A020704

Adjacent sequences:  A271940 A271941 A271942 * A271944 A271945 A271946

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 21 2016

STATUS

approved

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Last modified August 8 06:27 EDT 2020. Contains 336290 sequences. (Running on oeis4.)