More Identities On The Tribonacci 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
In this paper, we use a simple method to derive di¤erent recurrence relations on the Tribonacci numbers and their sums. By using the companion matrices and generating matrices, we get more identities on the Tribonacci numbers and their sums, which are more general than that given in literature [E. Kilic, Tribonacci Sequences with Certain Indices and Their Sum, Ars Combinatoria 86 (2008), 13-22.]. 1. Introduction The Tribonacci sequence is like the Fibonacci sequence, but instead of starting with two predetermined terms, the sequence starts with three predetermined terms and each term afterwards is the sum of the preceding three terms, that is Tn = Tn 1 + Tn 2 + Tn 3, n 3 (1.1) where T0 = T1 = 0; T2 = 1: The rst few tribonacci numbers are: 0; 0; 1; 1; 2; 4; 7; 13; 24; 44; 81; 149; 274; 504; 927; 1705; 3136; 5768; The tribonacci constant 1+ 3 p 19+3 p 33+ 3 p 19 3 p 33 3 is the ratio toward which adjacent tribonacci numbers tend. It is a root of the polynomial x x x 1, approximately 1.83929 , and also satis es the equation x 2x+1 = 0. In [1], the author derives new recurrence relations and generating matrices for the sums of usual Tribonacci numbers fSng and 4n subscripted Tribonacci numbers fT4ng, and their sums fS4ng, where Sn = Pn k=0 Tk. In this paper, we intend to give the more identities on the Tribonacci numbers fTn+wg, arbitrary subscripted Tribonacci numbers fTw(n+h)g, and their sums fSn+wg; fSw(n+h)g, where w and h are arbitrary positive integers. 2. Another Recurrence Relation By the recurrence (1.1), we have two expressions: Tn = Tn 1 + Tn 2 + Tn 3, and Tn 1 = Tn 2+Tn 3+Tn 4, subtract the second expression from 2000 Mathematics Subject Classi cation. 11B37, 15A36.
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 | 0.000 | 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