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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.17509 |
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| _version_ | 1866911525443207168 |
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| author | Pinn, Klaus |
| author_facet | Pinn, Klaus |
| contents | 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 $2^{αk}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17509 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Study of Meta-Fibonacci Integer Sequences by Continuous Self-Referential Functional Equations Pinn, Klaus Statistical Mechanics Dynamical Systems Chaotic Dynamics 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 $2^{αk}$. |
| title | Study of Meta-Fibonacci Integer Sequences by Continuous Self-Referential Functional Equations |
| topic | Statistical Mechanics Dynamical Systems Chaotic Dynamics |
| url | https://arxiv.org/abs/2603.17509 |