OFFSET
1,1
COMMENTS
Number of possible well-colored cycles on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.
LINKS
A.H.M. Smeets, Table of n, a(n) for n = 1..100
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.
FORMULA
E.g.f.: -log(1-2*x-4*x^2).
a(n) ~ sqrt(2*Pi/n)*(2*n*phi/e)^n. - Stefano Spezia, Jan 16 2024
MATHEMATICA
a[n_] := 2^n*LucasL[n, 1]*(n-1)!; Array[a, 19] (* or *)
nmax=19; CoefficientList[Series[-Log[1-2x-4x^2], {x, 0, nmax}], x]Range[0, nmax]! (* Stefano Spezia, Jan 15 2024 *)
PROG
(Python)
def A097632(n):
L0, L1, F, i = 1, 2, 2, 1
while i < n:
L0, L1, F, i = L0+L1, L0, 2*i*F, i+1
return L0*F # A.H.M. Smeets, Jan 15 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Ralf Stephan, Aug 17 2004
EXTENSIONS
Definition corrected by and a(18)-a(19) from Stefano Spezia, Jan 15 2024
STATUS
approved