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A273345 Number of levels in all bargraphs having semiperimeter n (n>=2). A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step. 4
0, 1, 2, 7, 23, 75, 245, 801, 2622, 8595, 28215, 92751, 305304, 1006207, 3320071, 10966741, 36261414, 120010103, 397528422, 1317860989, 4372180109, 14515485973, 48222552640, 160300772873, 533176676911, 1774359032599, 5907894024527, 19680307851415, 65588436120988, 218679463049627 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

LINKS

Table of n, a(n) for n=2..31.

A. Blecher, C. Brennan, and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9, 2015, 297-310.

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

a(n) = Sum(k*A273344(n,k), k>=0).

G.f. g(z) = (1-z)^2 (1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2 sqrt((1-z)(1-3z-z^2-z^3))).

Conjecture: n*a(n) +2*(-3*n+4)*a(n-1) +(9*n-28)*a(n-2) +2*a(n-3) +(-n+16)*a(n-4) +2*(-n+7)*a(n-5) +(-n+8)*a(n-6)=0. - R. J. Mathar, Jun 02 2016

EXAMPLE

a(4) = 2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3]; they have 1, 0, 0, 1, 0 levels, respectively.

MAPLE

g := (1/2)*(1-z)^2*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/sqrt((1-z)*(1-3*z-z^2-z^3)): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 2 .. 42);

CROSSREFS

Cf. A082582, A273344.

Sequence in context: A077832 A030282 A291015 * A042575 A256030 A052924

Adjacent sequences:  A273342 A273343 A273344 * A273346 A273347 A273348

KEYWORD

nonn

AUTHOR

Emeric Deutsch, May 21 2016

STATUS

approved

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Last modified July 21 21:14 EDT 2019. Contains 325199 sequences. (Running on oeis4.)