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.

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

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))).

D-finite with recurrence 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