OFFSET
2,1
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.
Row n (n>=3) contains 2n-3 entries.
Sum of entries of row n is (F(n+1) - 1)(2F(n+1) - 1), where F(j)=A000045(j) are the Fibonacci numbers.
T(n,1) = 2*F(n+1)-2 = number of edges in the Fibonacci tree of order n.
Sum(k*T(n,k), k>=0) = A180567 (the Wiener index of the Fibonacci tree of order n).
REFERENCES
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
LINKS
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
FORMULA
The Wiener polynomial w(n,t) of the Fibonacci tree of order n satisfies the recurrence relation w(n,t)=w(n-1,t) + w(n-2,t) + t*r(n-1,t) + t*r(n-2,t) + t^2*r(n-1,t)*r(n-2,t), w(0,t)=w(1,t)=0, where r(n,t) is the generating polynomial of the nodes of the Fibonacci tree of order n with respect to the level of the nodes (for example, 1+2t for the tree /\; see A178522 and the Maple program).
EXAMPLE
T(2,2)=1 because in the Fibonacci tree of order 2, namely /\, there is only 1 pair of nodes at distance 2 (the two leaves).
Triangle starts:
2,1;
4,4,2;
8,10,8,6,4;
14,19,20,18,18,12,4;
MAPLE
G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 20 do r[n] := sort(coeff(Gser, z, n)) end do: w[0] := 0: w[1] := 0: for n from 2 to 15 do w[n] := sort(expand(w[n-1]+w[n-2]+t*r[n-1]+t*r[n-2]+t^2*r[n-1]*r[n-2])) end do: 2, 1; for n from 3 to 10 do seq(coeff(w[n], t, j), j = 1 .. 2*n-3) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Sep 14 2010
STATUS
approved