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 A180567 The Wiener index of the Fibonacci tree of order n. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. REFERENCES Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. LINKS FORMULA a(n)=Sum(k*A180566(n,k), k>=0). 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] EXAMPLE a(2)=4 because in the tree /\ we have 3 distances: 1, 1, and 2. MAPLE 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); CROSSREFS Cf. A180566 Sequence in context: A059227 A081103 A005777 * A152392 A001563 A094304 Adjacent sequences:  A180564 A180565 A180566 * A180568 A180569 A180570 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 14 2010 STATUS approved

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