A165910 Wiener indices of Fibonacci trees of order k.

1, 4, 18, 62, 210, 666, 2063, 6226, 18484, 54100, 156620, 449268, 1278981, 3617544, 10175590, 28485218, 79406350, 220536910, 610487875, 1684974790, 4638298536, 12737460744, 34902844728, 95449821672, 260554112425, 710056257196

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

Table of n, a(n) for n=1..26.

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

Sequence in context: A292465 A227162 A057414 * A212766 A100177 A083321

Adjacent sequences: A165907 A165908 A165909 * A165911 A165912 A165913

Keyword

nonn

Author

K.V.Iyer, K. R. Udaya Kumar Reddy, Sep 30 2009

Status

approved