Some properties of generalized <span class="math inline">\((k,t)\)</span>-Jacobsthal <span class="math inline">\(p\)</span>-sequences
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Bibliographic record
Abstract
<p>In this paper, we generalize the <span class="math inline">\(k\)</span>-Jacobsthal sequences and call them the generalized <span class="math inline">\((k,t)\)</span>-Jacobsthal <span class="math inline">\(p\)</span>-sequences. Also, we obtain combinatorial identities. Then, the generalized<span class="math inline">\((k,t)\)</span>-Jacobsthal <span class="math inline">\(p\)</span>-matrix is used to factorize the Pascal matrix. Finally, using the Riordan method, we obtain two factorizations of the Pascal matrix involving the generalized <span class="math inline">\((k,t)\)</span>-Jacobsthal <span class="math inline">\(p\)</span>-sequences.</p>
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.001 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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