

A178524


Triangle read by rows: T(n,k) is the number of leaves at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n1).


3



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, 0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2, 0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2, 0, 0, 0, 0, 0, 0, 1, 27, 105, 140, 81, 21, 2
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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 n1 and whose right subtree is the Fibonacci tree of order n2; 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(n2).
T(n,k) is the number of vertices in the Fibonacci cube G_{n1} 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, AddisonWesley, Reading, MA, 1998, p. 417.


LINKS

Table of n, a(n) for n=0..91.
A. Castro and M. Mollard, The eccentricity sequences of Fibonacci and Lucas cubes, Discrete Math., 312 (2012), 10251037. 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, 168178.
S. Klavzar, M. Mollard, Asymptotic Properties of Fibonacci Cubes and Lucas Cubes, Annals of Combinatorics, 18, 2014, 447457.


FORMULA

G.f.: G(t,z) = (1+zt*z) / (1t*zt*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+zt*z)/(1t*zt*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 .. n1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A000045, A067331, A178522, A004070
Sequence in context: A112177 A115723 A238160 * A212357 A114525 A127672
Adjacent sequences: A178521 A178522 A178523 * A178525 A178526 A178527


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Jun 15 2010


STATUS

approved



