A Problem on Generation Sets Containing Fibonacci Numbers
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
At the Sixteenth International Conference on Fibonacci Numbers and Their Applications the following problem was posed by Clark Kimberling:Let S be the set generated by these rules: Let 1 ∈ S and if x ∈ S, then 2x ∈ S and 1 — x ∈ S, so that S grows in generations: G1 = {1}, G2 = {0, 2}, G3 = {–1, 4}, ….Prove or disprove that each generation contains at least one Fibonacci number or its negative.In this paper we generalize the problem as follows. Let S be the set described above, S be a sequence and Ƥ the property that a generation contains a term of S or the negative of a term of S. We will show that when S is the Fibonacci sequence there are many generations that fail to have property Ƥ. Other sequences S will also be considered and shown to have at least one generation failing to have property Ƥ. The proportion of generations failing to have property Ƥ is also investigated.
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.
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 | 0.001 | 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) | 0.000 | 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