Binomial Transform of the Generalized Tribonacci Sequence
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Published
Oct 8, 2020
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26-55
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Y¨uksel Soykan
Department of Mathematics, Art and Science Faculty, Zonguldak B¨ulent Ecevit University, 67100, Zonguldak, Turkey
Abstract
In this paper, we define the binomial transform of the generalized Tribonacci sequence and as special cases, the binomial transform of the Tribonacci, Tribonacci-Lucas, Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas sequences will be introduced. We investigate their properties in details. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these binomial transforms. Moreover, we give some identities and matrices related with these binomial transforms.
Keywords:
Binomial transform, Tribonacci sequence, Tribonacci numbers, Tribonacci-Lucas sequence, Tribonacci-Lucas numbers, binomial transform of Tribonacci sequence, 3-step Fibonacci sequence.
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Original Research Article
References
Bruce I. A modified Tribonacci Sequence. The Fibonacci Quarterly. 1984;22(3):244-246.
Catalani M. Identities for Tribonacci-related sequences; 2002. Available:https://arxiv.org/pdf/math/0209179.pdf math/0209179
Choi E. Modular Tribonacci numbers by matrix method. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 2013;20(3):207-221. Available:https://doi.org/10.7468/jksmeb.2013.20.3.207
Elia M. Derived sequences. The Tribonacci recurrence and cubic forms. The Fibonacci Quarterly. 2001;39(2):107-115.
Er MC. Sums of Fibonacci numbers by matrix methods. Fibonacci Quart. 1984;22(3):204-207.
Lin PY. De Moivre-type identities for the Tribonacci numbers. The Fibonacci Quarterly. 1988;26:131-134.
Pethe S. Some identities for Tribonacci sequences. The Fibonacci Quarterly. 1988;26(2):144-151.
Scott A, Delaney T, Hoggatt Jr. V. The Tribonacci sequence. The Fibonacci Quarterly.1977;15(3):193-200.
Shannon A. Tribonacci numbers and Pascal’s pyramid. The Fibonacci Quarterly.1977;15(3):268275.
Soykan Y. Tribonacci and Tribonacci-lucas sedenions. Mathematics. 2019;7(1):74.Available:https://doi.org/10.3390/math7010074
Spickerman W. Binet’s formula for the Tribonacci sequence. The Fibonacci Quarterly.1982;20:118-120.
Yalavigi CC. Properties of Tribonacci numbers. The Fibonacci Quarterly. 1972;10(3):231-246.
Yilmaz N, Taskara N. Tribonacci and Tribonacci-lucas numbers via the determinants of special Matrices. Applied Mathematical Sciences. 2014;8(39):1947-1955. Available:https://doi.org/10.12988/ams.2014.4270
Howard FT, Saidak F. Zhou’s theory of constructing identities. Congress Numer. 2010;200:225-237.
Soykan Y. On four special cases of generalized Tribonacci sequence: Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas Sequences. Journal of Progressive Research in Mathematics. 2020;16(3):3056-3084.
Sloane NJA. The on-line Encyclopedia of Integer Sequences. Available:http://oeis.org/
Knuth DE. The art of computer programming 3. Reading, MA: Addison Wesley; 1973.
Gould HW. Series transformations for finding recurrences for sequences. The Fibonacci Quarterly. 1990;28(2):166-171.
Haukkanen P. Formal power series for binomial sums of sequences of numbers. The Fibonacci Quarterly. 1993;31(1):28-31.
Prodinger H. Some information about the binomial transform. The Fibonacci Quarterly.1994;32(5):412-415.
Spivey MZ. Combinatorial sums and finite differences. Discrete Math. 2007;307:3130-3146. Available:https://doi.org/10.1016/j.disc.2007.03.052
Barry P. On integer-sequence-based cnstructions of gneralized pascal triangles. Journal of Integer Sequences. 2006;9. Article 06.2.4.
Soykan Y. Simson identity of generalized m-step Fibonacci numbers. Int. J. Adv. Appl. Math. and Mech. 2019;7(2):45-56. ISSN: 2347-2529
Soykan Y. Summing formulas for generalized Tribonacci numbers. Universal Journal of Mathematics and Applications. 2020;3(1):1-11. DOI:https://doi.org/10.32323/ujma.637876
Soykan Y. Generalized Tribonacci numbers: Summing formulas. Int. J. Adv. Appl. Math. and Mech. 2020;7(3):57-76.
Soykan Y. On the sums of squares of generalized Tribonacci numbers: Closed formulas of ∑n k=0 xkWk2. Archives of Current Research International. 2020;20(4):22-47. DOI: 10.9734/ACRI/2020/v20i430187
Soykan Y. A closed formula for the sums of squares of generalized Tribonacci numbers. Journal of Progressive Research in Mathematics. 2020;16(2):2932-2941.
Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 1982;20(1):73-76
Catalani M. Identities for Tribonacci-related sequences; 2002. Available:https://arxiv.org/pdf/math/0209179.pdf math/0209179
Choi E. Modular Tribonacci numbers by matrix method. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 2013;20(3):207-221. Available:https://doi.org/10.7468/jksmeb.2013.20.3.207
Elia M. Derived sequences. The Tribonacci recurrence and cubic forms. The Fibonacci Quarterly. 2001;39(2):107-115.
Er MC. Sums of Fibonacci numbers by matrix methods. Fibonacci Quart. 1984;22(3):204-207.
Lin PY. De Moivre-type identities for the Tribonacci numbers. The Fibonacci Quarterly. 1988;26:131-134.
Pethe S. Some identities for Tribonacci sequences. The Fibonacci Quarterly. 1988;26(2):144-151.
Scott A, Delaney T, Hoggatt Jr. V. The Tribonacci sequence. The Fibonacci Quarterly.1977;15(3):193-200.
Shannon A. Tribonacci numbers and Pascal’s pyramid. The Fibonacci Quarterly.1977;15(3):268275.
Soykan Y. Tribonacci and Tribonacci-lucas sedenions. Mathematics. 2019;7(1):74.Available:https://doi.org/10.3390/math7010074
Spickerman W. Binet’s formula for the Tribonacci sequence. The Fibonacci Quarterly.1982;20:118-120.
Yalavigi CC. Properties of Tribonacci numbers. The Fibonacci Quarterly. 1972;10(3):231-246.
Yilmaz N, Taskara N. Tribonacci and Tribonacci-lucas numbers via the determinants of special Matrices. Applied Mathematical Sciences. 2014;8(39):1947-1955. Available:https://doi.org/10.12988/ams.2014.4270
Howard FT, Saidak F. Zhou’s theory of constructing identities. Congress Numer. 2010;200:225-237.
Soykan Y. On four special cases of generalized Tribonacci sequence: Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas Sequences. Journal of Progressive Research in Mathematics. 2020;16(3):3056-3084.
Sloane NJA. The on-line Encyclopedia of Integer Sequences. Available:http://oeis.org/
Knuth DE. The art of computer programming 3. Reading, MA: Addison Wesley; 1973.
Gould HW. Series transformations for finding recurrences for sequences. The Fibonacci Quarterly. 1990;28(2):166-171.
Haukkanen P. Formal power series for binomial sums of sequences of numbers. The Fibonacci Quarterly. 1993;31(1):28-31.
Prodinger H. Some information about the binomial transform. The Fibonacci Quarterly.1994;32(5):412-415.
Spivey MZ. Combinatorial sums and finite differences. Discrete Math. 2007;307:3130-3146. Available:https://doi.org/10.1016/j.disc.2007.03.052
Barry P. On integer-sequence-based cnstructions of gneralized pascal triangles. Journal of Integer Sequences. 2006;9. Article 06.2.4.
Soykan Y. Simson identity of generalized m-step Fibonacci numbers. Int. J. Adv. Appl. Math. and Mech. 2019;7(2):45-56. ISSN: 2347-2529
Soykan Y. Summing formulas for generalized Tribonacci numbers. Universal Journal of Mathematics and Applications. 2020;3(1):1-11. DOI:https://doi.org/10.32323/ujma.637876
Soykan Y. Generalized Tribonacci numbers: Summing formulas. Int. J. Adv. Appl. Math. and Mech. 2020;7(3):57-76.
Soykan Y. On the sums of squares of generalized Tribonacci numbers: Closed formulas of ∑n k=0 xkWk2. Archives of Current Research International. 2020;20(4):22-47. DOI: 10.9734/ACRI/2020/v20i430187
Soykan Y. A closed formula for the sums of squares of generalized Tribonacci numbers. Journal of Progressive Research in Mathematics. 2020;16(2):2932-2941.
Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quart. 1982;20(1):73-76