

A192019


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


1



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
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OFFSET

2,2


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(k1) as its left subtree and f(k2) as its right subtree. See the Iyer & Reddy references.


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=2..27.
B. E. Sagan, YN. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959969.
K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wiener index of binomial trees and Fibonacci trees, arXiv:0910.4432 [cs.DM], 2009.


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(n1,t) + w(n2,t) + t*r(n1,t) + t*r(n2) + t^2*r(n1,t)*r(n2,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^43*x^2+4*x+1)/((x+1)^2*(x^23*x+1)^2*(x^2+x1)^2). [Colin Barker, Nov 17 2012]


EXAMPLE

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.


MAPLE

G := z/((1z)*(1t*zt*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[n1]+w[n2]+t*r[n1]+t*r[n2]+t^2*r[n1]*r[n2])) end do: seq(subs(t = 1, diff(w[n], t)), n = 2 .. 27);


CROSSREFS

Cf. A192018.
Sequence in context: A261648 A086462 A201830 * A201210 A003207 A095687
Adjacent sequences: A192016 A192017 A192018 * A192020 A192021 A192022


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 21 2011


STATUS

approved



