A273905 - OEIS

%I #20 Mar 08 2023 06:08:48

%S 1,2,3,5,9,15,27,46,83,143,259,450,817,1429,2599,4570,8323,14698,

%T 26797,47491,86659,154042,281287,501283,915907,1635835,2990383,

%U 5351138,9786369,17541671,32092959,57610988,105435607,189521640,346950321,624389105

%N Number of symmetric bargraphs having semiperimeter n (n>=2).

%H Alois P. Heinz, <a href="/A273905/b273905.txt">Table of n, a(n) for n = 2..2000</a>

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016

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

%F Conjecture D-finite with recurrence (n+1)*a(n) +2*(-1)*a(n-1) +2*(-2*n+3)*a(n-2) +2*(n-6)*a(n-4) +6*(1)*a(n-5) -2*a(n-6) +(n-9)*a(n-8)=0. - _R. J. Mathar_, Jul 22 2022

%F a(n) ~ sqrt(2*r*(2-3*r)) * (25 + 18*r + 13*r^2) * (1 + r + r^2)^n / (22*sqrt(Pi*n)), where r = A192918. - _Vaclav Kotesovec_, Mar 08 2023

%e a(4) = 3; indeed, the corresponding compositions are [3],[2,2],[1,1,1].

%e a(6) = 9; indeed, the corresponding compositions are [5],[4,4],[1,3,1],[2,3,2],[2,1,2],[3,3,3],[2,2,2,2],[1,2,2,1],[1,1,1,1,1].

%p Q := sqrt((1-z^2)*(1-z-z^2-z^3)*(1+z-z^2+z^3)): g := (1/2)*(1+z)*(z^4+2*z^3+2*z^2-1+Q)/(z*(1-z-z^2-z^3)): gser := series(g, z = 0, 42): seq(coeff(gser,z,n), n=2..37);

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n<9, [$0..3, 5, 9, 15, 27]

%p       [n], (2*a(n-1) +(4*n-6)*a(n-2) -(2*n-12)*a(n-4)

%p       -6*a(n-5) +2*a(n-6) -(n-9)*a(n-8))/ (n+1))

%p     end:

%p seq(a(n), n=2..40);  # _Alois P. Heinz_, Jun 24 2016

%t a[2]=1; a[3]=2; a[4]=3; a[5]=5; a[6]=9; a[7]=15; a[8]=27; a[n_ /; n>8] := a[n] = ((9-n)*a[n-8] + 2*a[n-6] - 6*a[n-5] + (12-2*n)*a[n-4] + (4*n-6)*a[n-2] + 2*a[n-1])/(n+1); Table[a[n], {n, 2, 40}] (* _Jean-François Alcover_, Dec 02 2016, after _Alois P. Heinz_ *)

%K nonn

%O 2,2

%A _Emeric Deutsch_ and _Sergi Elizalde_, Jun 23 2016