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The \(q\)-Calkin-Wilf tree. (English) Zbl 1232.05025

Summary: We define a \(q\)-analogue of the Calkin-Wilf tree and the Calkin-Wilf sequence. We show that the \(n\)th term \(f(n;q)\) of the \(q\)-analogue of the Calkin-Wilf sequence is the generating function for the number of hyperbinary expansions of \(n\) according to the number of powers that are used exactly twice. We also present formulae for branches within the \(q\)-analogue of the Calkin-Wilf tree and predecessors and successors of terms in the \(q\)-analogue of the Calkin-Wilf sequence.

MSC:

05A30 \(q\)-calculus and related topics
05C05 Trees
05C07 Vertex degrees
05C10 Planar graphs; geometric and topological aspects of graph theory
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References:

[1] Aigner, M.; Ziegler, G. M., Proofs from the Book (2009), Springer · Zbl 0978.00002
[2] Alkauskas, G.; Steuding, J., Statistical properties of the Calkin-Wilf tree: real an \(p\)-adic distribution
[3] Bates, B.; Bunder, M.; Tognetti, K., Linking the Calkin-Wilf and Stern-Brocot trees, European J. Combin., 31, 1637-1661 (2010) · Zbl 1209.05045
[4] Calkin, N.; Wilf, H. S., Recounting the rationals, Amer. Math. Monthly, 107, 360-363 (2000) · Zbl 0983.11009
[5] Cole, A. J.; Davie, A. J.T., A game based on the Euclidean algorithm and a winning strategy for it, Math. Gaz., 53, 354-357 (1969) · Zbl 0186.25303
[6] Hofmann, S.; Schuster, G.; Steuding, J., Euclid, Calkin & Wilf — Playing with rationals, Elem. Math., 63, 109-117 (2008) · Zbl 1193.91033
[7] Knuth, D. E., Problem 10906, American Mathematical Monthly; solution by Moshe Newman, Amer. Math. Monthly, 110, 642-643 (2003)
[8] Reznick, B., Regularity properties of the Stern enumeration of the rationals, J. Integer Seq., 11 (2008), Article 08.4.1, 17 pp · Zbl 1204.11027
[9] Stern, M. A., Über eine zahlentheoretische Funktion, J. Reine Angew. Math., 55, 193-220 (1858) · ERAM 055.1457cj
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