On the Behavior of a Family of Meta-Fibonacci Sequences

Callaghan, Joseph H.; Chew, John J.; Tanny, Stephen M.

doi:10.1137/s0895480103421397

Cited by 13 publications

(11 citation statements)

References 7 publications

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“…Notice that the second sequence is the same as the first except for the extra repetition of the powers of 2. This kind of kinship among the solutions to certain closely related meta-Fibonacci recursions derived from (1.1) is well-known (see, for example, [4], [5], and [14]); it will also be a feature of some of the new results for other special cases of (1.1) that we introduce later in this paper.…”

Section: Introductionmentioning

confidence: 62%

“…Once again the results are intriguing: with just one exception, namely, (1, 3 : 3, 5) 4 , the only recursions with an apparent solution have the parameters (s, j : j + s, 2j) or (s, 1 : 2 + s, 3), where s > 0 and j > 0; further, all of these solutions are slow growing sequences! 5 The recursion (s, j : s + j, 2j) is a natural generalization of the recursion (0, j : j, 2j) identified in the first set of calculations with "shift" parameter s; 6 in addition, it is a 4 Subsequently, additional investigation over a greater range of values for the parameters has lead to the identification of another exception, the recursion (2, 5 : 4, 7). Both of these exceptional recursions have essentially the same slow solution, namely, the ceiling function for n 2 .…”

Section: A Brief Empirical Interludementioning

confidence: 99%

“…In recent years various special cases of (1.1), together with alternative sets of initial conditions, have been analyzed. See, for example, [1], [2], [3], [4], [5], [6], [7], [11], [14], [15], [18], [20]. These contributions illustrate the very wide range of behavior that can be exhibited by meta-Fibonacci sequences that derive from (1.1).…”

Section: Introductionmentioning

confidence: 99%

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Trees and Meta-Fibonacci Sequences

Isgur

Reiss

Tanny

2009

Electron. J. Combin.

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For $k>1$ and nonnegative integer parameters $a_p, b_p$, $p = 1..k$, we analyze the solutions to the meta-Fibonacci recursion $C(n)=\sum_{p=1}^k C(n-a_p-C(n-b_p))$, where the parameters $a_p, b_p$, $p = 1..k$ satisfy a specific constraint. For $k=2$ we present compelling empirical evidence that solutions exist only for two particular families of parameters; special cases of the recursions so defined include the Conolly recursion and all of its generalizations that have been studied to date. We show that the solutions for all the recursions defined by the parameters in these families have a natural combinatorial interpretation: they count the number of labels on the leaves of certain infinite labeled trees, where the number of labels on each node in the tree is determined by the parameters. This combinatorial interpretation enables us to determine various new results concerning these sequences, including a closed form, and to derive asymptotic estimates. Our results broadly generalize and unify recent findings of this type relating to certain of these meta-Fibonacci sequences. At the same time they indicate the potential for developing an analogous counting interpretation for many other meta-Fibonacci recursions specified by the same recursion for $C(n)$ with other sets of parameters.

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Section: Introductionmentioning

confidence: 62%

Section: A Brief Empirical Interludementioning

confidence: 99%

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Trees and Meta-Fibonacci Sequences

Isgur

Reiss

Tanny

2009

Electron. J. Combin.

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“…The unary tree K that generates Golomb's recursive sequence. 3 The step function property implies that g(n) has a closed form, namely, g(n) = ⌊ ⌊ √ 8n⌋+1 2 ⌋. See [5].…”

Section: Further Applicationsmentioning

confidence: 99%

Solving Non-Homogeneous Nested Recursions Using Trees

2013

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Abstract. The solutions to certain nested recursions, such as Conolly's C(n) = C(n − C(n − 1)) + C(n − 1 − C(n − 2)), with initial conditions C(1) = 1, C(2) = 2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a k-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions.

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“…See Table 2 for the first twenty terms of the sequence L(n), which can readily be computed from T (20) in Figure 1. 5 Our strategy is to show that for all n, L(n) = H(n) and that L(n) is the Conolly sequence. Note that last label in T (20) L(n) has the same 5 initial values as H(n), namely, 1, 2, 2, 3, 4.…”

Section: Order 1 Conolly-like Recursionsmentioning

confidence: 99%

Nested Recurrence Relations with Conolly-like Solutions

Erickson¹,

Isgur²,

Jackson³

et al. 2012

SIAM J. Discrete Math.

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A nondecreasing sequence of positive integers is (α, β)-Conolly, or Conollylike for short, if for every positive integer m the number of times that m occurs in the sequence is α + βr m , where r m is 1 plus the 2-adic valuation of m. A recurrence relation is (α, β)-Conolly if it has an (α, β)-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the formA(n − a ij )) with appropriate initial conditions. For any fixed integers k and p 1 , p 2 , . . . , p k we prove that there are only finitely many pairs (α, β) for which A(n) can be (α, β)-Conolly. For the case where α = 0 and β = 1, we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence H(n) = H(n − H(n − 2)) + H(n − 3 − H(n − 5)) also has the Conolly sequence as a solution. When k = 2 and p 1 = p 2 , we construct an example of an (α, β)-Conolly recursion for every possible (α, β) pair, thereby providing the first examples of nested recursions with p i > 1 whose solutions are completely understood. Finally, in the case where k = 2 and p 1 = p 2 , we provide an if and only if condition for a given nested recurrence A(n) to be (α, 0)-Conolly by proving a very general ceiling function identity.

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On the Behavior of a Family of Meta-Fibonacci Sequences

Cited by 13 publications

References 7 publications

Trees and Meta-Fibonacci Sequences

Trees and Meta-Fibonacci Sequences

Solving Non-Homogeneous Nested Recursions Using Trees

Nested Recurrence Relations with Conolly-like Solutions

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