OFFSET
0,4
COMMENTS
A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
Sum of entries in row n is the Fibonacci number F(n+1) (A000045(n+1)).
Sum(k*T(n,k),k>=0)=A067331(n-2).
T(n,k) is the number of vertices in the Fibonacci cube G_{n-1} that have eccentricity k (see Klavzar and Mollard reference). - Michel Mollard, Aug 20 2014
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
LINKS
A. Castro and M. Mollard, The eccentricity sequences of Fibonacci and Lucas cubes, Discrete Math., 312 (2012), 1025-1037. See Table 1. [From N. J. A. Sloane, Mar 22 2012]
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
S. Klavzar, M. Mollard, Asymptotic Properties of Fibonacci Cubes and Lucas Cubes, Annals of Combinatorics, 18, 2014, 447-457.
FORMULA
G.f.: G(t,z) = (1+z-t*z) / (1-t*z-t*z^2).
EXAMPLE
Triangle starts:
1,
1,
0, 2,
0, 1, 2,
0, 0, 3, 2,
0, 0, 1, 5, 2,
0, 0, 0, 4, 7, 2,
0, 0, 0, 1, 9, 9, 2,
0, 0, 0, 0, 5, 16, 11, 2,
0, 0, 0, 0, 1, 14, 25, 13, 2,
0, 0, 0, 0, 0, 6, 30, 36, 15, 2,
MAPLE
G := (1+z-t*z)/(1-t*z-t*z^2): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 15 2010
STATUS
approved