Emerson, Nathaniel D. A family of meta-Fibonacci sequences defined by variable-order recursions. (English) Zbl 1122.11010 J. Integer Seq. 9, No. 1, Article 06.1.8, 21 p. (2006). Author’s abstract: 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. Reviewer: Krassimir Atanassov (Sofia) Cited in 1 Document MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B37 Recurrences Software:OEIS PDFBibTeX XMLCite \textit{N. D. Emerson}, J. Integer Seq. 9, No. 1, Article 06.1.8, 21 p. (2006; Zbl 1122.11010) Full Text: arXiv EuDML EMIS Online Encyclopedia of Integer Sequences: Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1. Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1. Narayana-Zidek-Capell numbers: a(n) = 1 for n <= 2. Otherwise a(2n) = 2a(2n-1), a(2n+1) = 2a(2n) - a(n). Hofstadter-Conway \(10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1\) Hofstadter Q-sequence: a(1) = a(2) = 1; a(n) = a(n-a(n-1)) + a(n-a(n-2)) for n > 2. A well-behaved cousin of the Hofstadter sequence: a(n) = a(n - 1 - a(n-1)) + a(n - 2 - a(n-2)) for n > 2 with a(0) = a(1) = a(2) = 1. Array F(k, n) read by descending antidiagonals: k-generalized Fibonacci numbers in row k >= 1, starting (0, 1, 1, ...), for column n >= 0. a(0)=0; for n >= 1, a(n) = a(n-1-A023416(n)) + A000120(n). a(0)=1; a(1)=1; for n >= 2, a(n) = a(A023416(n)) + a(A000120(n)). a(0)=1; a(1)=1; for n >= 2, a(n) = a(n-A000120(n)) + a(n-1-A023416(n))