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 A180566 Triangle read by rows: T(n,k) is the number of unordered pairs of nodes at distance k in the Fibonacci tree of order n (entries in row n are the coefficients of the corresponding Wiener polynomial). 3
 2, 1, 4, 4, 2, 8, 10, 8, 6, 4, 14, 19, 20, 18, 18, 12, 4, 24, 34, 40, 44, 48, 46, 40, 20, 4, 40, 58, 72, 88, 106, 114, 122, 112, 76, 28, 4, 66, 97, 124, 160, 208, 242, 284, 310, 308, 244, 128, 36, 4, 108, 160, 208, 276, 376, 466, 576, 686, 782, 812, 720, 472, 196, 44, 4, 176 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 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. Row n (n>=3) contains 2n-3 entries. Sum of entries of row n is (F(n+1) - 1)(2F(n+1) - 1), where F(j)=A000045(j) are the Fibonacci numbers. T(n,1) = 2*F(n+1)-2 = number of edges in the Fibonacci tree of order n. Sum(k*T(n,k), k>=0) = A180567 (the Wiener index of the Fibonacci tree of order n). REFERENCES D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. LINKS Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178. B. E. Sagan, Y-N. Yeh and P. Zhang, The Wiener Polynomial of a Graph, Internat. J. of Quantum Chem., 60, 1996, 959-969. FORMULA 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 and the Maple program). EXAMPLE T(2,2)=1 because in the Fibonacci tree of order 2, namely /\, there is only 1 pair of nodes at distance 2 (the two leaves). Triangle starts: 2,1; 4,4,2; 8,10,8,6,4; 14,19,20,18,18,12,4; MAPLE G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 20 do r[n] := sort(coeff(Gser, z, n)) end do: w := 0: w := 0: for n from 2 to 15 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: 2, 1; for n from 3 to 10 do seq(coeff(w[n], t, j), j = 1 .. 2*n-3) end do; # yields sequence in triangular form CROSSREFS Cf. A178522, A180567 Sequence in context: A135366 A247248 A192017 * A051289 A090802 A129159 Adjacent sequences:  A180563 A180564 A180565 * A180567 A180568 A180569 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Sep 14 2010 STATUS approved

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Last modified November 18 22:33 EST 2019. Contains 329305 sequences. (Running on oeis4.)