OFFSET
1,3
COMMENTS
a(n) is the number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and horizontal steps H(s) = (s,0) for every positive integer s, such that H(s) is colored by means of F(s) colors, where F(s) is the s-th Fibonacci number.
LINKS
R. De Castro, A. Ramírez, and J. Ramírez, Applications in enumerative combinatorics of infinite weighted automata and graphs, arXiv:1310.2449, (2013).
FORMULA
G.f.: (1-2*z-z^2-sqrt((1-2*z-z^2)^2-4*z^2*(1-z-z^2)^2))/(2*z^2*(1-z-z^2)).
D-finite with recurrence: n*(n+2)*a(n) = (7*n^2 + 5*n - 1)*a(n-1) - (14*n^2 - 8*n + 1)*a(n-2) + (3*n^2 - 12*n + 23)*a(n-3) + (n+1)*(13*n - 38)*a(n-4) - 4*(n-4)*n*a(n-5) - 4*(n-5)*(n+1)*a(n-6). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ sqrt(663+161*sqrt(17)) * ((3+sqrt(17))/2)^(n-3/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 15 2014
MATHEMATICA
Rest[CoefficientList[Series[(1-2*x-x^2-Sqrt[(1-2*x-x^2)^2-4*x^2*(1-x-x^2)^2])/(2*x^2*(1-x-x^2)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 15 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Oct 10 2013
STATUS
approved