OFFSET
0,3
COMMENTS
a(n) = number of Motzkin paths (A001006) of length n that contain no consecutive UFU.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Jean-Luc Baril and José Luis Ramírez, Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises, arXiv:2301.10449 [math.CO], 2023.
FORMULA
G.f.: (1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)).
EXAMPLE
a(5) = 20 because, of the 21 Motzkin paths of length 5, only UFUDD contains an occurrence of UFU.
MATHEMATICA
CoefficientList[Series[(1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6)^(1/2))/(2*x^2*(1 - x + x^2)), {x, 0, 30}], x] (* Michael De Vlieger, Jan 31 2023 *)
PROG
(PARI) seq(n)={Vec((1 - x - x^3 - (1 - 2*x - 3*x^2 + 2*x^3 - 2*x^4 + x^6 + O(x^3*x^n))^(1/2))/(2*x^2*(1 - x + x^2)))} \\ Andrew Howroyd, Nov 05 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
David Callan, Jul 16 2004
EXTENSIONS
Terms a(23) and beyond from Andrew Howroyd, Nov 05 2019
STATUS
approved