Study of Meta-Fibonacci Integer Sequences by Continuous Self-Referential Functional Equations

Abstract

I propose and investigate the use of continuous functional equations for the study of meta-Fibonacci integer sequences. This exploratory study includes three sequences with quite different behavior: Conway's famous sequence $A(n)= A(A(n-1))+A(n-A(n-1))$, the sequence $D(n)= D(D(n-1))+D(n-1-D(n-2))$ introduced by the present author more than 25 years ago, and Hofstadter's well-known $Q(n)= Q(n-Q(n-1))+Q(n-Q(n-2))$. The sequences are studied in their equivalent detrended forms $(a,d,q)(n)=2\,(A,D,Q)(n)-n$. For $a(n)$ and $d(n)$, a highly symmetric functional equation admits exact continuous solutions that nicely model the global behavior (backbone) of the sequences. For the Hofstadter sequence, a continuous functional model is developed that leads to a random matrix approach for the generation and study of fractal solutions. Two remarkable properties of the Q-sequence are reproduced by the model: the anomalous scaling of the generation length, which scales $\sim (2-η)^k$, and the anomalous amplitude growth that scales like $2^{αk}$.

Figures (8)

• Figure 1: $A(n)$, $D(n)$, and $Q(n)$ for $n \leq 1024$.
• Figure 2: $a(n)$, $d(n)$, and $q(n)$ for $n \leq 1024$.
• Figure 3: Solution of the CME with triangle start function for $K=0,...,6$.
• Figure 4: The solution of the CME with triangle start function, for $K=10$ and $K=11$, together with their large-$K$-approximations according to Eq. (\ref{['approx_large_K']}).
• Figure 5: $d(n)$ and $a(n)$, $d(n)$, together with their piecewise linear backbone functions $n \leq 2048$. The first figure shows the $d$-backbone in some more detail for $n\leq 256$.

Theorems & Definitions (14)

• Conjecture 1
• Conjecture 2
• Lemma 1: Flip Lemma
• proof
• Lemma 2: Shift Lemma
• proof
• Lemma 3: Composition Lemma
• proof
• Theorem 1: Triangle Theorem
• proof : Proof of Triangle Theorem