On the Behavior of a Variant of Hofstadter's Q-Sequence
Why this work is in the frame
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Bibliographic record
Abstract
We completely solve the meta-Fibonacci recursion V (n) = V (n − V (n − 1)) + V (n − V (n − 4)), a variant of Hofstadter’s meta-Fibonacci Q-sequence. For the initial conditions V (1) = V (2) = V (3) = V (4) = 1 we prove that the sequence V (n) is monotone, with successive terms increasing by 0 or 1, so the sequence hits every positive integer. We demonstrate certain special structural properties and fascinating periodicities of the associated frequency sequence (the number of times V (n) hits each positive integer) that make possible an iterative computation of V (n) for any value of n. Further, we derive a natural partition of the V -sequence into blocks of consecutive terms (“generations”) with the property that terms in one block determine the terms in the next. We conclude by examining all the other sets of four initial conditions for which this metaFibonacci recursion has a solution; we prove that in each case the resulting sequence is essentially the same as the one with initial conditions all ones.
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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 | 0.003 | 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 | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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