On the Behavior of a Family of Meta-Fibonacci Sequences
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Bibliographic record
Abstract
A family of meta-Fibonacci sequences is defined by the k-term recursion $$ T_{a,k}(n) :=\sum_{i=0}^{k-1}T_{a,k}({n-i-a-T_{a,k}(n-i-1)}), \quad n>a+k,\,k\ge2, $$ with initial conditions $T_{a,k}(n)=1$ for $1\le n \le a+k$. Some partial results are obtained for $a\ge 0$ and $k>1$. The case a=0 and k odd is analyzed in detail, giving a complete characterization of its structure and behavior, marking the first time that such a parametric family of meta-Fibonacci sequences has been solved. This behavior is considerably more complex than that of the more familiar Conolly sequence (a=0, k=2). Various properties are derived: for example, a certain difference of summands turns out to consist of palindromic subsequences, and the mean values of the functions on these subsequences are computed. Conjectures are made concerning the still more complex behavior of a=0 and even k > 2.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it