Some Sequences Resembling Hofstadter's

Nested Recurrence Relations with Conolly-like Solutions

“…Aside from the original Conolly recursion 0; 1 : 1; 2 [1,2] our detailed computer search identified only one candidate, namely, 0; 2 : 3; 5 [1,2,2,3,4]. We now prove that the Conolly sequence satisfies this recursion.…”

“…Let A(n) satisfy the recursion 1; 1 : 3; 3 [0, 0, 0, 0], which generates a sequence of all zeroes, so A(n) n converges to 0. Let B(n) satisfy the recursion 1; 1 : 3; 3 [1,1,1,2], that is, the same recursion but with different initial conditions. The solution B(n) is the slow sequence in which all numbers that are not powers of two appear twice, and all numbers that are powers of two appear three times, so B(n) n converges to 1 2 (for a proof of these facts about B(n) see [4]).…”

“…(1.2) For convenience we use this notation to refer to both the recursion and its solution sequence even in the situation where we are uncertain that a solution exists. For example Hofstadter's Q-sequence ( [10]), defined by Q(1) = Q(2) = 1 and Q(n) = Q(n−Q(n−1))+Q(n−Q(n−2)) for n > 2 is written Q = 0; 1 : 0; 2 [1,1]; Q is a famous example where it is not known whether or not a solution exists.…”

“…Our empirical work indicated that many combinations of parameters and initial conditions yield a solution sequence reminiscent of the well-known Conolly sequence [6]. This sequence is the solution to the nested (also called self-referencing or meta-Fibonacci) recurrence relation C(n) = 0; 1 : 1; 2 [1,2]. We say it is slowly growing or slow because it has the property that successive differences are either 0 or 1 and it tends to infinity.…”

“…Correspondingly, we say that a recurrence relation is Conolly-like if it has a Conolly-like solution sequence and is (α, β)-Conolly when the solution is (α, β)-Conolly. 1 In this paper, we exclude from consideration the special case where −α = β > 0, as the resulting sequence would not be slow.…”