A274495 The length of the longest initial sequence of the form UHUH..., summed over all bargraphs having semiperimeter n (n>=2).
2, 3, 9, 23, 62, 171, 482, 1384, 4036, 11924, 35619, 107407, 326521, 999675, 3079634, 9539366, 29693294, 92831327, 291366477, 917765199, 2900217452, 9192097510, 29213057684, 93073003438, 297215560553, 951144390092, 3049877146281, 9797605279905
Offset: 2
Keywords
Examples
a(4) = 9 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that the sum of the lengths of their longest initial sequence of the form UHUH... is 2+4+1+1+1.
Links
- M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
- Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016
Programs
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Maple
Q := sqrt((1-z)*(1-3*z-z^2-z^3)): g := (((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)*(1/2))/(z*(1-z)): gser := series(g, z = 0, 38): seq(coeff(gser, z, n), n = 2 .. 34);
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Mathematica
terms = 28; g[z_] = (((1-z)(1 - 4z^2 - 3z^3 - 2z^4) - (1 + z - z^2 - 2z^3)*Q)(1/2))/(z (1-z)) /. Q -> Sqrt[(1-z)(1 - 3z - z^2 - z^3)]; Drop[CoefficientList[g[z] + O[z]^(terms+2), z], 2] (* Jean-François Alcover, Aug 07 2018 *)
Formula
G.f.: g(z) = ((1-z)*(1-4*z^2-3*z^3-2*z^4)-(1+z-z^2-2*z^3)*Q)/(2*z*(1-z)), where Q = sqrt((1-z)*(1-3*z-z^2-z^3)).
a(n) = Sum_{k>=1} k*A274494(n,k).
D-finite with recurrence -(n+1)*(19*n-44)*a(n) +n*(43*n-65)*a(n-1) +2*(47*n^2-289*n+342)*a(n-2) +2*(-33*n^2+170*n-61)*a(n-3) +(-19*n^2+87*n+22)*a(n-4) -(33*n-31)*(n-8)*a(n-5)=0. - R. J. Mathar, Jul 22 2022