A125267 Number of Motzkin paths with no peaks and with level steps at height 0 having three colors except that consecutive level steps at height 0 must have different colors.
1, 3, 6, 13, 30, 71, 171, 417, 1026, 2542, 6333, 15849, 39813, 100329, 253518, 642117, 1629726, 4143857, 10553511, 26916426, 68739015, 175752268, 449846001, 1152528593, 2955487605, 7585165701, 19481930556, 50073211027, 128784497466, 331426205715, 853409723277
Offset: 0
Keywords
Examples
a(3) = 13 since there are 12 = 3*2*2 paths that stay at level 0 and one path ULD that goes above level 0.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Crossrefs
Cf. A004148.
Programs
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Mathematica
CoefficientList[Series[(((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)*(1 + x))/(1 - x*((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 10 2017 *)
Formula
G.f.: (g(x)*(1+x))/(1-x*g(x)) where g(x)=((1-x+x^2)-sqrt((1-x+x^2)^2-4x^2))/(2*x^2).
Conjecture: -(n+1)*(n-2)*a(n) +2*(n^2-n-3)*a(n-1) +(n^2-3*n+8)*a(n-2) +2*(n^2-5*n+3)*a(n-3) -(n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 17 2016
a(n) ~ 5^(1/4) * phi^(2*n+1) / sqrt(Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022
Comments