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A192019
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The Wiener index of the binary Fibonacci tree of order n.
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1
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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
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2,2
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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FORMULA
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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]
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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