Nested Recurrence Relations with Conolly-like Solutions
Bibliographic record
Abstract
A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer m the number of times that m occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is 1 plus the 2-adic valuation of m. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers k and $p_1,p_2,\ldots, p_k$ we prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$, we provide a bijective proof using labeled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$, we construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$, we provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".