Stephen M. Tanny

A well-behaved cousin of the Hofstadter sequence

Abstract Very little is known about the Hofstadter sequence Q ( n ) defined by Q (1) = Q (2) = 1 and Q ( n ) = Q ( n − Q ( n −1)) + Q ( n − Q ( n −2)), n > 2. A seemingly close relative is the sequence T ( n ) given by T ( n ) = T ( n −1− T ( n −1)) + T ( n −2− T ( n−2 ), n > 2 with T (0) = T (1) = T (2) = 1. In sharp contrast to the ‘chaotic’ behaviour of Q ( n ), T ( n ) behaves in a completely predictable fashion which is characterized precisely. In particular, T ( n ) is monotonic and hits every positive integer.

Permutations and successions

Abstract Let π = ( π (1), π (2),…, π ( n )) be a permutation on {1, 2, …, n }. A succession (respectively, ∗ -succession) in π is any pair π ( i ), π ( i + 1), where π ( i + 1) = π ( i ) + 1 (respectively, π ( i + 1) ≡ π ( i ) + 1 (mod n )), i = 1, 2, …, n − 1. Let R ( n , k ) (respectively, R ∗ (n, k) ) be the number of permutations with k successions (respectively, ∗ -successions). In this note we determine R ( n , k ) and R ∗ (n, k) . In addition, these notions are generalized to the case of circular permutations, where analogous results are developed.

Periodicities of solutions of the generalized lyness recursion

The mapping of the real plane into itself is used to examine the periodicity of the solutions to the generalized Lyness recursion when c is fixed, we show that this mapping is periodic with given period independent of (x,y)if and only if c = 0 or c = 1. Certain cubic curves remain invariant under Tc and some of these are loci of periodic points for different periods. We characterize the loci and values of c which yield periodic sequences with periods 1 through 10. We conclude with a conjecture concerning the occurrence of periodic sequences with higher periodicities.

On the Behavior of a Family of Meta-Fibonacci Sequences

A family of meta-Fibonacci sequences is defined by the k-term recursion

Nested Recurrence Relations with Conolly-like Solutions

T_{a,k}(n) :=\sum_{i=0}^{k-1}T_{a,k}({n-i-a-T_{a,k}(n-i-1)}), \quad n>a+k,\,k\ge2,

Constructing New Families of Nested Recursions with Slow Solutions

with initial conditions

Spot-Based Generations for Meta-Fibonacci Sequences

T_{a,k}(n)=1

A combinatorial approach for solving certain nested recursions with non-slow solutions

for