OFFSET
1,2
COMMENTS
The Fibonacci trees T(f(k)) of order k is defined as follows: 1. T(f(-1)) and T(f(0)) each consist of a single node. 2. For k >= 1, T(f(k)) is built from copies of T(f(k-1)) and T(f(k-2)) by connecting (by an edge) T(f(k-2)) as the rightmost child of the root of T(f(k-1)).
REFERENCES
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of Binomial trees and Fibonacci trees, Int'l. J. Math. Engin. with Comp., Accepted for publication, Sept. 2009.
LINKS
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wienerindex of binomial trees and Fibonacci trees, arXiv:0910.4432
Index entries for linear recurrences with constant coefficients, signature (5, -3, -14, 10, 14, -5, -3, 1).
FORMULA
The Wiener index W(T(f(k))) of the Fibonacci tree T(f(k)) satisfies the following recurrence: W(T(f(k))) = W(T(f(k-1))) + W(T(f(k-2))) + F(k+1) D(T(f), (k-2)) + F(k) D(T(f), (k-1)) + F(k+1) F(k), where D(T(f), k) = (1/5) (k F(k+2) + (k+2) F(k)) and F(k) is the k-th Fibonacci number.
D(T(f),k) = A001629(k+1). Conjecture: G.f. x*(1-x+x^2-2*x^3)/( (1-x^2-x) * (1+x)^2 * (x^2-3*x+1)^2 ). [From R. J. Mathar, Apr 19 2010]
EXAMPLE
W(T(1)) = 1 because T(1) is a single edge. W(T(2)) = 4 because T(2) is a path on three vertices.
CROSSREFS
KEYWORD
nonn
AUTHOR
K.V.Iyer, K. R. Udaya Kumar Reddy, Sep 30 2009
STATUS
approved