A192019 The Wiener index of the binary Fibonacci tree of order n.

1, 10, 50, 214, 802, 2802, 9275, 29580, 91668, 277924, 828092, 2433140, 7067885, 20337318, 58054534, 164602410, 463990190, 1301338150, 3633753815, 10107239160, 28016346216, 77419909800, 213349801560, 586471432104, 1608485221177, 4402406713762

Offset: 2

Author

Emeric Deutsch, Jun 21 2011

Keywords

• nonn

Comments

The binary Fibonacci trees f(k) of order k are rooted binary trees defined as follows: 1. f(0) has no nodes and f(1) consists of a single node. 2. For k>=2, f(k) is constructed from a root with f(k-1) as its left subtree and f(k-2) as its right subtree. See the Iyer & Reddy references.

Examples

			a(3)=10 because the binary Fibonacci tree of order 3 is basically the path graph A - B - R - C and we have 3 distances equal to 1 (AB, BR, RC), 2 distances equal to 2 (AR and BC) and 1 distance equal to 3 (AC); 3*1 + 2*2 + 1*3 = 10.
		

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

• B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969.
• K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.

Crossrefs

Cf. A192018.

Programs

Maple

G := z/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 30)): for n to 27 do r[n] := sort(coeff(Gser, z, n)) end do: w[1] := 0: w[2] := t: for n from 3 to 27 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: seq(subs(t = 1, diff(w[n], t)), n = 2 .. 27);

Formula

a(n) = Sum_{k>=1} k*A192018(n,k).

The Wiener index of a connected graph is the derivative of the Wiener polynomial W(t) of the graph, evaluated at t=1. The Wiener polynomial w(n,t) of the binary 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^2*r(n-1,t)*r(n-2,t), w(1,t)=0, w(2,t)=t, where r(n,t) is the generating polynomial of the nodes of the binary Fibonacci tree f(n) with respect to the level of the nodes (for example, r(2,t) = 1 + t for the tree / ; see A004070 and the Maple program).

Empirical G.f.: x^2*(x^4-3*x^2+4*x+1)/((x+1)^2*(x^2-3*x+1)^2*(x^2+x-1)^2). [Colin Barker, Nov 17 2012]