On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences
Abstract
:1. Introduction
- (i)
- has a dominant root, say , which is its only positive root, withfor all . In particular, tends to as .
- (ii)
- has a negative root (which is unique) only when k is even.
- (iii)
- All the roots of are simple roots.
- (iv)
- is a strictly decreasing sequence.
2. Auxiliary Results
3. The Proof of the Main Theorem
3.1. Proof of Item (i)
3.2. Proof of Item (ii)
3.3. Proof of Item (iii)
3.4. Proof of Item (iv)
4. Applications of Our Results
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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k | k | k | ||||||
---|---|---|---|---|---|---|---|---|
2 | 2.414213562 | 2.414213562 | 18 | 1.163910449 | 1.342997170 | 34 | 1.097462321 | 1.246182982 |
3 | 1.839286755 | 2.000000000 | 19 | 1.156780140 | 1.333333333 | 35 | 1.095189446 | 1.242535625 |
4 | 1.618033989 | 1.816496581 | 20 | 1.150313062 | 1.324442842 | 36 | 1.093031609 | 1.239045722 |
5 | 1.497094049 | 1.707106781 | 21 | 1.144417473 | 1.316227766 | 37 | 1.090979976 | 1.235702260 |
6 | 1.419632763 | 1.632455532 | 22 | 1.139018098 | 1.308606700 | 38 | 1.089026607 | 1.232495277 |
7 | 1.365254707 | 1.577350269 | 23 | 1.134052557 | 1.301511345 | 39 | 1.087164353 | 1.229415734 |
8 | 1.324717957 | 1.534522484 | 24 | 1.129468689 | 1.294883912 | 40 | 1.085386751 | 1.226455407 |
9 | 1.293188036 | 1.500000000 | 25 | 1.125222520 | 1.288675135 | 41 | 1.083687949 | 1.223606798 |
10 | 1.267874775 | 1.471404521 | 26 | 1.121276701 | 1.282842712 | 42 | 1.082062631 | 1.220863052 |
11 | 1.247047862 | 1.447213595 | 27 | 1.117599293 | 1.277350098 | 43 | 1.080505957 | 1.218217890 |
12 | 1.229573607 | 1.426401433 | 28 | 1.114162811 | 1.272165527 | 44 | 1.079013511 | 1.215665546 |
13 | 1.214676212 | 1.408248290 | 29 | 1.110943467 | 1.267261242 | 45 | 1.077581254 | 1.213200716 |
14 | 1.201805729 | 1.392232270 | 30 | 1.107920561 | 1.262612866 | 46 | 1.076205487 | 1.210818511 |
15 | 1.190560750 | 1.377964473 | 31 | 1.105075990 | 1.258198890 | 47 | 1.074882811 | 1.208514414 |
16 | 1.180640991 | 1.365148372 | 32 | 1.102393848 | 1.254000254 | 48 | 1.073610101 | 1.206284249 |
17 | 1.171817047 | 1.353553391 | 33 | 1.099860103 | 1.250000000 | 49 | 1.072384476 | 1.204124145 |
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Trojovský, P. On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences. Mathematics 2020, 8, 1387. https://doi.org/10.3390/math8081387
Trojovský P. On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences. Mathematics. 2020; 8(8):1387. https://doi.org/10.3390/math8081387
Chicago/Turabian StyleTrojovský, Pavel. 2020. "On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences" Mathematics 8, no. 8: 1387. https://doi.org/10.3390/math8081387