On a generalization of the Pell sequence

Jhon J. Bravo; Jose L. Herrera; Florian Luca

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 2, page 199-213
  • ISSN: 0862-7959

Abstract

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The Pell sequence ( P n ) n = 0 ∞ is the second order linear recurrence defined by P n = 2 P 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.

How to cite

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Bravo, Jhon J., Herrera, Jose L., and Luca, Florian. "On a generalization of the Pell sequence." Mathematica Bohemica 146.2 (2021): 199-213. <http://eudml.org/doc/298178>.

@article{Bravo2021,
abstract = {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.},
author = {Bravo, Jhon J., Herrera, Jose L., Luca, Florian},
journal = {Mathematica Bohemica},
keywords = {generalized Fibonacci number; generalized Pell number; recurrence sequence},
language = {eng},
number = {2},
pages = {199-213},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a generalization of the Pell sequence},
url = {http://eudml.org/doc/298178},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Bravo, Jhon J.
AU - Herrera, Jose L.
AU - Luca, Florian
TI - On a generalization of the Pell sequence
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 2
SP - 199
EP - 213
AB - 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.
LA - eng
KW - generalized Fibonacci number; generalized Pell number; recurrence sequence
UR - http://eudml.org/doc/298178
ER -

References

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  1. Bicknell, M., A primer on the Pell sequence and related sequences, Fibonacci Q. 13 (1975), 345-349. (1975) Zbl0319.10013MR0387173
  2. Bravo, J. J., Luca, F., 10.5486/PMD.2013.5390, Publ. Math. 82 (2013), 623-639. (2013) Zbl1274.11035MR3066434DOI10.5486/PMD.2013.5390
  3. Brent, R. P., 10.2307/2153583, Math. Comput. 63 (1994), 389-401. (1994) Zbl0809.11083MR1216256DOI10.2307/2153583
  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. (2014) Zbl1360.11031MR3181762
  5. Horadam, A. F., Applications of modified Pell numbers to representations, Ulam Q. 3 (1995), 35-53. (1995) Zbl0874.11023MR1368399
  6. Kalman, D., Generalized Fibonacci numbers by matrix methods, Fibonacci Q. 20 (1982), 73-76. (1982) Zbl0472.10016MR0660765
  7. Kiliç, E., On the usual Fibonacci and generalized order- k Pell numbers, Ars Comb. 88 (2008), 33-45. (2008) Zbl1224.11024MR2426404
  8. Kiliç, E., 10.1016/j.ejc.2007.03.004, Eur. J. Comb. 29 (2008), 701-711. (2008) Zbl1138.11004MR2397350DOI10.1016/j.ejc.2007.03.004
  9. Kiliç, E., Taşci, D., The linear algebra of the Pell matrix, Bol. Soc. Mat. Mex., III. Ser. 11 (2005), 163-174. (2005) Zbl1092.05004MR2207722
  10. Kiliç, E., Taşci, D., 10.11650/twjm/1500404581, Taiwanese J. Math. 10 (2006), 1661-1670. (2006) Zbl1123.11005MR2275152DOI10.11650/twjm/1500404581
  11. Koshy, T., 10.1002/9781118033067, Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley, New York (2001). (2001) Zbl0984.11010MR1855020DOI10.1002/9781118033067
  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. (2001) Zbl0989.11008MR1829526
  13. Marques, D., On k -generalized Fibonacci numbers with only one distinct digit, Util. Math. 98 (2015), 23-31. (2015) Zbl1369.11014MR3410879
  14. E. P. Miles, Jr., 10.2307/2308649, Am. Math. Mon. 67 (1960), 745-752. (1960) Zbl0103.27203MR0123521DOI10.2307/2308649
  15. Muskat, J. B., 10.2307/2152961, Math. Comput. 61 (1993), 365-372. (1993) Zbl0781.11006MR1192974DOI10.2307/2152961
  16. Wolfram, D. A., Solving generalized Fibonacci recurrences, Fibonacci Q. 36 (1998), 129-145. (1998) Zbl0911.11014MR1622060
  17. Wu, Z., Zhang, H., On the reciprocal sums of higher-order sequences, Adv. Difference Equ. 2013 (2013), Paper No. 189, 8 pages 9999DOI99999 10.1186/1687-1847-2013-189 . (2013) Zbl1390.11042MR3084191

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