0,4

The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

G. C. Greubel, Table of n, a(n) for n = 0..1000

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

a(n) = A182891(n,0).

G.f.: G(z) =1/( z^2+sqrt((1+z+z^2)*(1-3*z+z^2)) ).

a(n) ~ sqrt(360 + 161*sqrt(5)) * ((3 + sqrt(5))/2)^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2016. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 6) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021

Conjecture: n*a(n) +(n-2)*a(n-1) +2*(-9*n+16)*a(n-2) +5*(2*n-5)*a(n-3) +(10*n-33) *a(n-4) +2*(26*n-109)*a(n-5) +(13*n-37)*a(n-6) +(13*n-63) *a(n-7) +10*(-n+7) *a(n-8)=0. - R. J. Mathar, Jun 14 2016

a(3)=3. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; three of them, namely ud, du, and hhh, have no H-steps at level 0.

G:=1/(z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=series(G, z=0, 35): seq(coeff(Gser, z, n), n=0..30);

CoefficientList[Series[1/(x^2+Sqrt[(1+x+x^2)(1-3x+x^2)]), {x, 0, 30}], x] (* Harvey P. Dale, Aug 25 2012 *)

(PARI) z='z+O('z^50); Vec(1/( z^2+sqrt((1+z+z^2)*(1-3*z+z^2)) )) \\ G. C. Greubel, Mar 26 2017

Cf. A182891.

Sequence in context: A077970 A338852 A174284 * A124696 A081669 A086821

Adjacent sequences: A182889 A182890 A182891 * A182893 A182894 A182895

Emeric Deutsch, Dec 12 2010

approved