The Combinatorics of Certain k-ary Meta-Fibonacci Sequences

This paper is concerned with a family of k -ary meta-Fibonacci sequences described by the recurrence relation a(n) = Sigma(k)(i =1) a(n-i-(s-1)-a(n-i)), where s may be any integer, positive or negative. If s >= 0, then the initial conditions are a(n) = 1 for 1 <= n <= s+1 and a(n) = n-s for s+1 < n <= s+k . If s <= 0, then the initial conditions are a ( n ) = n for 1 ? n ? k ( - s + 1). We show that these sequences arise as the solutions of two natural counting problems: The numbe r of leaves at the largest level in certain infinite k -ary trees, and (for s >= 0) certain restricted compositions of an integer. For this family of generalized meta-Fibonacci se quences and two families of related sequences we derive combinatorial bijections, o rdinary generating functions, recurrence relations, and asymptotics (a(n) similar to n(k-1)/k ). We also show that these sequences are related to a self-describing sequence of Clo itre and Sloane.