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%I #7 Jul 28 2015 09:20:27
%S 1,4,18,62,210,666,2063,6226,18484,54100,156620,449268,1278981,
%T 3617544,10175590,28485218,79406350,220536910,610487875,1684974790,
%U 4638298536,12737460744,34902844728,95449821672,260554112425,710056257196
%N Wiener indices of Fibonacci trees of order k.
%C 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)).
%D 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.
%H K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, <a href="http://arxiv.org/abs/0910.4432">Wienerindex of binomial trees and Fibonacci trees</a>, arXiv:0910.4432
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (5, -3, -14, 10, 14, -5, -3, 1).
%F 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.
%F 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]
%e W(T(1)) = 1 because T(1) is a single edge. W(T(2)) = 4 because T(2) is a path on three vertices.
%K nonn
%O 1,2
%A _K.V.Iyer_, K. R. Udaya Kumar Reddy, Sep 30 2009
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