| Title:
|
On a generalization of the Pell sequence (English) |
| Author:
|
Bravo, Jhon J. |
| Author:
|
Herrera, Jose L. |
| Author:
|
Luca, Florian |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
146 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
199-213 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The Pell sequence $(P_n)_{n=0}^{\infty }$ is the second order linear recurrence defined by $P_n=2P_{n-1}+P_{n-2}$ with initial conditions $P_0=0$ and $P_1=1$. In this paper, we investigate a generalization of the Pell sequence called the $k$-generalized Pell sequence which is generated by a recurrence relation of a higher order. We present recurrence relations, the generalized Binet formula and different arithmetic properties for the above family of sequences. Some interesting identities involving the Fibonacci and generalized Pell numbers are also deduced. (English) |
| Keyword:
|
generalized Fibonacci number |
| Keyword:
|
generalized Pell number |
| Keyword:
|
recurrence sequence |
| MSC:
|
11B37 |
| MSC:
|
11B39 |
| DOI:
|
10.21136/MB.2020.0098-19 |
| . |
| Date available:
|
2021-05-20T13:55:02Z |
| Last updated:
|
2021-06-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148932 |
| . |
| Reference:
|
[1] Bicknell, M.: A primer on the Pell sequence and related sequences.Fibonacci Q. 13 (1975), 345-349. Zbl 0319.10013, MR 0387173 |
| Reference:
|
[2] Bravo, J. J., Luca, F.: On a conjecture about repdigits in $k$-generalized Fibonacci sequences.Publ. Math. 82 (2013), 623-639. Zbl 1274.11035, MR 3066434, 10.5486/PMD.2013.5390 |
| Reference:
|
[3] Brent, R. P.: On the periods of generalized Fibonacci recurrences.Math. Comput. 63 (1994), 389-401. Zbl 0809.11083, MR 1216256, 10.2307/2153583 |
| Reference:
|
[4] Dresden, G. P. B., Du, Z.: A simplified Binet formula for $k$-generalized Fibonacci numbers.J. Integer Seq. 17 (2014), Article No. 14.4.7, 9 pages. Zbl 1360.11031, MR 3181762 |
| Reference:
|
[5] Horadam, A. F.: Applications of modified Pell numbers to representations.Ulam Q. 3 (1995), 35-53. Zbl 0874.11023, MR 1368399 |
| Reference:
|
[6] Kalman, D.: Generalized Fibonacci numbers by matrix methods.Fibonacci Q. 20 (1982), 73-76. Zbl 0472.10016, MR 0660765 |
| Reference:
|
[7] Kiliç, E.: On the usual Fibonacci and generalized order-$k$ Pell numbers.Ars Comb. 88 (2008), 33-45. Zbl 1224.11024, MR 2426404 |
| Reference:
|
[8] Kiliç, E.: The Binet formula, sums and representations of generalized Fibonacci $p$-numbers.Eur. J. Comb. 29 (2008), 701-711. Zbl 1138.11004, MR 2397350, 10.1016/j.ejc.2007.03.004 |
| Reference:
|
[9] Kiliç, E., Taşci, D.: The linear algebra of the Pell matrix.Bol. Soc. Mat. Mex., III. Ser. 11 (2005), 163-174. Zbl 1092.05004, MR 2207722 |
| Reference:
|
[10] Kiliç, E., Taşci, D.: The generalized Binet formula, representation and sums of the generalized order-$k$ Pell numbers.Taiwanese J. Math. 10 (2006), 1661-1670. Zbl 1123.11005, MR 2275152, 10.11650/twjm/1500404581 |
| Reference:
|
[11] Koshy, T.: Fibonacci and Lucas Numbers with Applications. Vol. I.Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, New York (2001). Zbl 0984.11010, MR 1855020, 10.1002/9781118033067 |
| Reference:
|
[12] Lee, G.-Y., Lee, S.-G., Kim, J.-S., Shin, H.-K.: The Binet formula and representations of $k$-generalized Fibonacci numbers.Fibonacci Q. 39 (2001), 158-164. Zbl 0989.11008, MR 1829526 |
| Reference:
|
[13] Marques, D.: On $k$-generalized Fibonacci numbers with only one distinct digit.Util. Math. 98 (2015), 23-31. Zbl 1369.11014, MR 3410879 |
| Reference:
|
[14] E. P. Miles, Jr.: Generalized Fibonacci numbers and associated matrices.Am. Math. Mon. 67 (1960), 745-752. Zbl 0103.27203, MR 0123521, 10.2307/2308649 |
| Reference:
|
[15] Muskat, J. B.: Generalized Fibonacci and Lucas sequences and rootfinding methods.Math. Comput. 61 (1993), 365-372. Zbl 0781.11006, MR 1192974, 10.2307/2152961 |
| Reference:
|
[16] Wolfram, D. A.: Solving generalized Fibonacci recurrences.Fibonacci Q. 36 (1998), 129-145. Zbl 0911.11014, MR 1622060 |
| Reference:
|
[17] Wu, Z., Zhang, H.: On the reciprocal sums of higher-order sequences.Adv. Difference Equ. 2013 (2013), Paper No. 189, 8 pages \99999DOI99999 10.1186/1687-1847-2013-189 \vfil. Zbl 1390.11042, MR 3084191 |
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