OFFSET
0,5
COMMENTS
a(n)=A190170(n,0).
FORMULA
G.f. G=G(z) is obtained by elimitaing S from the equations G=1+zG+z^2*G(S-1-z) and S=1+zS+z^2*S(S-1).
From Vaclav Kotesovec, May 29 2022: (Start)
G.f.: 2/(1 + sqrt((1 + (-3 + x)*x)*(1 + x + x^2)) + x*(-1 + x + 2*x^2)).
a(n) ~ 5^(1/4) * phi^(2*n+6) / (18 * sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. (End)
Conjecture D-finite with recurrence (n+2)*a(n) +(-n+1)*a(n-1) +2*(-2*n-1)*a(n-2) +9*a(n-3) +(-n+1)*a(n-4) -9*a(n-5) +2*(-2*n+5)*a(n-6) +(-n+1)*a(n-7) +(n-4)*a(n-8)=0. - R. J. Mathar, Jul 22 2022
EXAMPLE
a(4)=2 because we have HHHH and UHHD.
MAPLE
p1 := G-1-z*G-z^2*G*(S-1-z): p2 := S-1-z*S-z^2*S*(S-1): r := resultant(p1, p2, S): g := RootOf(r, G): Gser := simplify(series(g, z = 0, 40)): seq(coeff(Gser, z, n), n = 0 .. 33);
MATHEMATICA
CoefficientList[Series[2/(1 + Sqrt[(1 + (-3 + x)*x)*(1 + x + x^2)] + x*(-1 + x + 2*x^2)), {x, 0, 40}], x] (* Vaclav Kotesovec, May 29 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 06 2011
STATUS
approved