Geodesic
A family of meta-Fibonacci sequences defined by variable-order recursions
Journal of integer sequences, Tome 9 (2006) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We define a family of meta-Fibonacci sequences. For each sequence in the family, the order of the of the defining recursion at the $n^{th}$ stage is a variable $r(n)$, and the $n^{th}$ term is the sum of the previous $r(n)$ terms. Given a sequence of real numbers that satisfies some conditions on growth, there is a meta-Fibonacci sequence in the family that grows at the same rate as the given sequence. In particular, the growth rate of these sequences can be exponential, polynomial, or logarithmic. However, the possible asymptotic limits of such a sequence are restricted to a class of exponential functions. We give upper and lower bounds for the terms of any such sequence, which depend only on $r(n)$. The Narayana-Zidek-Capell sequence is a member of this family. We show that it converges asymptotically.
Classification : 11B37, 11B39, 11B99
Keywords: meta-Fibonacci, hofstadter sequence, narayana-zidek-capell sequence 
@article{JIS_2006__9_1_a1,
     author = {Emerson, Nathaniel D.},
     title = {A family of {meta-Fibonacci} sequences defined by variable-order recursions},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2006__9_1_a1/}
}
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Emerson, Nathaniel D. A family of meta-Fibonacci sequences defined by variable-order recursions. Journal of integer sequences, Tome 9 (2006) no. 1. http://geodesic.mathdoc.fr/item/JIS_2006__9_1_a1/