A273720 Number of horizontal steps in the peaks of all bargraphs having semiperimeter n (n>=2).

1, 3, 8, 21, 57, 162, 479, 1458, 4528, 14259, 45349, 145289, 468121, 1515128, 4922145, 16040310, 52411294, 171646085, 563266323, 1851661113, 6096654978, 20101681834, 66362538332, 219336702948, 725692113292, 2403295565913, 7966021263923, 26425616887971

Offset: 2

Author

Emeric Deutsch, Jun 01 2016

Keywords

• nonn

Examples

			a(4) = 8 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 they have 3,1,1,2,1 horizontal steps in their peaks.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..1000
• 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. A075126, A082582, A273719.

Programs

Maple

g := (1/2)*z^2*(1-2*z+2*z^2-2*z^3+z^4+Q)/((1-z)^2*Q): Q := sqrt((1-z)^5*(1-3*z-z^2-z^3)): gser := series(g, z = 0, 35): seq(coeff(gser, z, n), n = 2 .. 32);
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0$2, 1, 3, 8, 21][n+1],
     ((2*(3*n-7))*(2*n-9)*a(n-1) -(254-155*n+22*n^2)*a(n-2)
      +(2*(101-58*n+8*n^2))*a(n-3) -(86-47*n+6*n^2)*a(n-4)
      +(2*(n-6))*(2*n-5)*a(n-5)-(n-6)*(2*n-5)*a(n-6))/
      ((n-2)*(2*n-9)))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 01 2016

Mathematica

a[n_] := a[n] = If[n<6, {0, 0, 1, 3, 8, 21}[[n+1]], ((2*(3*n-7))*(2*n - 9)*a[n-1] - (254 - 155*n + 22*n^2)*a[n-2] + (2*(101 - 58*n + 8*n^2))*a[n - 3] - (86 - 47*n + 6*n^2)*a[n-4] + (2*(n-6))*(2*n - 5)*a[n-5] - (n-6)*(2*n - 5)*a[n-6])/((n-2)*(2*n - 9))]; Table[a[n], {n, 2, 40}] (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)

Formula

G.f.: g(z) = z^2*(1-2*z+2*z^2-2*z^3+z^4+Q)/(2*Q*(1-z)^2), where Q = sqrt((1-z)^5*(1-3*z-z^2-z^3)).

a(n) = Sum(k*A273719(n,k), k>=1).

a(n) = ((2*(3*n-7))*(2*n-9)*a(n-1) -(254-155*n+22*n^2)*a(n-2) +(2*(101 -58*n +8*n^2))*a(n-3) -(86-47*n+6*n^2)*a(n-4) +(2*(n-6))*(2*n-5)*a(n-5) -(n-6)*(2*n-5)*a(n-6))/((n-2)*(2*n-9)) for n>=6. - Alois P. Heinz, Jun 01 2016