Binomial Transform of the Generalized Third Order Pell Sequence
DOI:
https://doi.org/10.26713/cma.v12i1.1371Keywords:
Binomial transform, Third order Pell sequence, Third order Pell numbers, Third order Pell-Lucas sequence, Third order Pell-Lucas numbersAbstract
In this paper, we define the binomial transform of the generalized third order Pell sequence and as special cases, the binomial transform of the third order Pell, third Order Pell-Lucas and modified third order Pell sequences will be introduced. We investigate their properties in details.
Downloads
References
P. Barry, On integer-sequence-based constructions of generalized pascal triangles, Journal of Integer Sequences 9 (2006), Article 06.2.4, http://emis.math.tifr.res.in/journals/JIS/VOL9/Barry/barry91.pdf.
I. Bruce, A modified Tribonacci sequence, The Fibonacci Quarterly 22(3) (1984), 244 – 246, https://www.mathstat.dal.ca/FQ/Scanned/22-3/bruce.pdf.
M. Catalani, Identities for tribonacci-related sequences, 6 pages, arXiv preprint, (2002) https://arxiv.org/pdf/math/0209179.pdfmath/0209179.
E. Choi, Modular tribonacci numbers by matrix method, Korean Society of Mathematical Education: The Pure and Applied Mathematics 20(3) (2013), 207 – 221, DOI: 10.7468/jksmeb.2013.20.3.207.
M. Elia, Derived sequences, the tribonacci recurrence and cubic forms, The Fibonacci Quarterly 39(2) (2001), 107 – 115, https://www.fq.math.ca/Scanned/39-2/elia.pdf.
M. C. Er, Sums of fibonacci numbers by matrix methods, The Fibonacci Quarterly 22(3) (1984), 204 – 207, https://www.fq.math.ca/Scanned/22-3/er.pdf.
H. W. Gould, Series transformations for finding recurrences for sequences, The Fibonacci Quarterly 28(2) (1990), 166 – 171, https://www.fq.math.ca/Scanned/28-2/gould.pdf.
P. Haukkanen, Formal power series for binomial sums of sequences of numbers, The Fibonacci Quarterly 31(1) (1993), 28 – 31, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.376.2491&rep=rep1&type=pdf.
F. T. Howard, F. Saidak, Zhou's theory of constructing identities, Congress Numer. 200 (2010), 225 – 237.
D. Kalman, Generalized fibonacci numbers by matrix methods, The Fibonacci Quarterly 20(1) (1982), 73 – 76, https://www.fq.math.ca/Scanned/20-1/kalman.pdf.
E. Kiliç and P. Stanica, A matrix approach for general higher order linear recurrences, Bulletin of the Malaysian Mathematical Sciences Society (2) 34(1) (2011), 51 – 67, http://math.usm.my/bulletin/pdf/v34n1/v34n1p5.pdf.
D. E. Knuth., The Art of Computer Programming: Sorting and Searching, Vol. 3, 2nd edition, Addison Wesley, Reading, MA (1998), https://doc.lagout.org/science/0_Computer%20Science/2_Algorithms/The%20Art%20of%20Computer%20Programming%20%28vol.%203_%20Sorting%20and%20Searching%29%20%282nd%20ed.%29%20%5BKnuth%201998-05-04%5D.pdf.
P.-Y. Lin, De Moivre-type identities for the tribonacci numbers, The Fibonacci Quarterly 26 (1988), 131 – 134, DOI: 10.1007/978-94-011-3586-3_24.
S. Pethe, Some identities for tribonacci sequences, The Fibonacci Quarterly 26(2) (1988), 144 – 151, https://www.mathstat.dal.ca/FQ/Scanned/26-2/pethe.pdf.
H. Prodinger, Some information about the binomial transform, The Fibonacci Quarterly 32(5) (1994), 412 – 415, https://www.fq.math.ca/Scanned/32-5/prodinger.pdf.
A. Scott, T. Delaney, V. Hoggatt Jr., The tribonacci sequence, The Fibonacci Quarterly 15(3) (1977), 193 – 200, https://www.fq.math.ca/Issues/15-3.pdf.
N. J. A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/.[18] A. Shannon, Tribonacci numbers and Pascal's pyramid, The Fibonacci Quarterly 15(3) (1977), 268, https://www.fq.math.ca/Issues/15-3.pdf.
Y. Soykan, Simson identity of generalized m-step fibonacci numbers, International Journal of Advances in Applied Mathematics and Mechanics 7(2) (2019), 45 – 56, http://www.ijaamm.com/uploads/2/1/4/8/21481830/v7n2p4_45-56.pdf.
Y. Soykan, Tribonacci and tribonacci-Lucas sedenions., Mathematics 7(1) (2019), 74, DOI: 10.3390/math7010074.
Y. Soykan, On generalized third-order Pell numbers, Asian Journal of Advanced Research and Reports 6(1) (2019), 1 – 18, Article no. AJARR.51635, DOI: 10.9734/ajarr/2019/v6i130144.
Y. Soykan, Summing formulas for generalized tribonacci numbers, Universal Journal of Mathematics and Applications 3(1) (2020), 1 – 11, DOI: 10.32323/ujma.637876.
W. Spickerman, Binet's formula for the Tribonacci sequence, The Fibonacci Quarterly 20 (1982), 118 – 120, https://www.fq.math.ca/Scanned/20-2/spickerman.pdf.
M. Z. Spivey, Combinatorial sums and finite differences, Discrete Math. 307 (2007), 3130 – 3146, DOI: 10.1016/j.disc.2007.03.052.
C. C. Yalavigi, Properties of tribonacci numbers, The Fibonacci Quarterly 10(3) (1972), 231 – 246, https://www.fq.math.ca/Issues/10-3.pdf.
N. Yilmaz and N. Taskara, Tribonacci and tribonacci-Lucas numbers via the determinants of special matrices, Applied Mathematical Sciences 8(39) (2014), 1947 – 1955, DOI: 10.12988/ams.2014.4270.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.