cp's OEIS Frontend

A180567 The Wiener index of the Fibonacci tree of order n.

%I A180567 #10 Jan 05 2025 19:51:39
%S A180567 0,0,4,18,96,374,1380,4696,15336,48318,148448,446890,1324104,3872656,
%T A180567 11206764,32143818,91509120,258855006,728211180,2038815272,5684262480,
%U A180567 15789141750,43712852544,120663667538,332191809936,912339490464
%N A180567 The Wiener index of the Fibonacci tree of order n.
%C A180567 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.
%C A180567 The Wiener index of a connected graph is the sum of the distances between all unordered pairs of nodes in the graph.
%D A180567 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A180567 Y. Horibe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
%H A180567 B. E. Sagan, Y-N. Yeh and P. Zhang, <a href="http://dx.doi.org/10.1002/(SICI)1097-461X(1996)60:5&lt;959::AID-QUA2&gt;3.0.CO;2-W">The Wiener Polynomial of a Graph</a>, Internat. J. of Quantum Chem., 60, 1996, 959-969.
%F A180567 a(n) = Sum(k*A180566(n,k), k>=0).
%F A180567 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).
%F A180567 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]
%e A180567 a(2)=4 because in the tree /\ we have 3 distances: 1, 1, and 2.
%p A180567 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);
%Y A180567 Cf. A180566
%K A180567 nonn
%O A180567 0,3
%A A180567 _Emeric Deutsch_, Sep 14 2010