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A165910
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Wiener indices of Fibonacci trees of order k.
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2
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1, 4, 18, 62, 210, 666, 2063, 6226, 18484, 54100, 156620, 449268, 1278981, 3617544, 10175590, 28485218, 79406350, 220536910, 610487875, 1684974790, 4638298536, 12737460744, 34902844728, 95449821672, 260554112425, 710056257196
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The Fibonacci trees T(f(k)) of order k is defined as follows: 1. T(f(-1)) and T(f(0)) each consist of a single node. 2. For k >= 1, T(f(k)) is built from copies of T(f(k-1)) and T(f(k-2)) by connecting (by an edge) T(f(k-2)) as the rightmost child of the root of T(f(k-1)).
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REFERENCES
| 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
| K. Viswanathan Iyer and K. R. Udaya Kumar Reddy, Wienerindex of binomial trees and Fibonacci trees, arXiv:0910.4432
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FORMULA
| The Wiener index W(T(f(k))) of the Fibonacci tree T(f(k)) satisfies the following recurrence: W(T(f(k))) = W(T(f(k-1))) + W(T(f(k-2))) + F(k+1) D(T(f), (k-2)) + F(k) D(T(f), (k-1)) + F(k+1) F(k), where D(T(f), k) = (1/5) (k F(k+2) + (k+2) F(k)) and F(k) is the k-th Fibonacci number.
D(T(f),k) = A001629(k+1). Conjecture: G.f. x*(1-x+x^2-2*x^3)/( (1-x^2-x) * (1+x)^2 * (x^2-3*x+1)^2 ). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 19 2010]
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EXAMPLE
| W(T(1)) = 1 because T(1) is a single edge. W(T(2)) = 4 because T(2) is a path on three vertices.
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CROSSREFS
| Sequence in context: A192069 A073373 A057414 * A100177 A083321 A022728
Adjacent sequences: A165907 A165908 A165909 * A165911 A165912 A165913
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KEYWORD
| nonn
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AUTHOR
| Kailasam Viswanathan Iyer, K.R.Udaya Kumar Reddy (kvi(AT)nitt.edu), Sep 30 2009
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