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A180567
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The Wiener index of the Fibonacci tree of order n.
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1
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0, 0, 4, 18, 96, 374, 1380, 4696, 15336, 48318, 148448, 446890, 1324104, 3872656, 11206764, 32143818, 91509120, 258855006, 728211180, 2038815272, 5684262480, 15789141750, 43712852544, 120663667538, 332191809936, 912339490464
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OFFSET
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0,3
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COMMENTS
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A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of nodes in the graph.
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REFERENCES
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D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
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LINKS
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FORMULA
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The Wiener polynomial w(n,t) of the 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) + t^2*r(n-1,t)*r(n-2,t), w(0,t)=w(1,t)=0, where r(n,t) is the generating polynomial of the nodes of the Fibonacci tree of order n with respect to the level of the nodes (for example, 1+2t for the tree /\; see A178522). The Wiener index is the derivative of w(n,t) with respect to t, evaluated at t=1 (see the Maple program).
Empirical G.f.: -2*x^2*(x^7-2*x^6-6*x^5+6*x^4+6*x^3-8*x^2+3*x-2)/((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(2)=4 because in the tree /\ we have 3 distances: 1, 1, and 2.
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MAPLE
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G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 38)): for n from 0 to 35 do r[n] := sort(coeff(Gser, z, n)) end do: w[0] := 0: w[1] := 0: for n from 2 to 30 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 = 0 .. 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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