Generalized Pell-Padovan Numbers
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Published
Jun 20, 2020
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8-28
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Yüksel Soykan
Department of Mathematics, Faculty of Art and Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey.
Abstract
In this paper, we investigate the generalized Pell-Padovan sequences and we deal with, in detail, four special cases, namely, Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell sequences. We present Binet’s formulas, generating functions, Simson formulas and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.
Keywords:
Pell-Padovan numbers, Pell-Perrin numbers, third order Fibonacci-Pell numbers, third order Lucas-Pell numbers.
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Original Research Article
References
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Catalani M. Identities for Tribonacci-related sequences; 2012. arXiv:math/0209179
Choi E. Modular Tribonacci numbers by matrix method. Journal of the Korean Society of Mathematical Education
Series B: Pure and Applied. Mathematics. 2013;20(3):207-221.
Elia M. Derived sequences, the Tribonacci recurrence and cubic forms. Fibonacci Quarterly. 2001;39(2):107-115.
Er MC. Sums of Fibonacci numbers by matrix methods. Fibonacci Quarterly. 1984;22(3):204-207.
Lin PY. De Moivre-type identities for the Tribonacci numbers. Fibonacci Quarterly. 1988;26:131-134.
Pethe S. Some identities for Tribonacci sequences. Fibonacci Quarterly. 1988;26(2):144-151.
Scott A, Delaney T, Hoggatt Jr. V. The Tribonacci sequence. Fibonacci Quarterly. 1977;15(3):193-200.
Shannon A. Tribonacci numbers and Pascal’s pyramid. Fibonacci Quarterly. 1977;15(3):268&275.
Soykan Y. Tribonacci and Tribonacci-lucas sedenions. Mathematics. 2019;7(1):74.
Spickerman W. Binet’s formula for the Tribonacci sequence. Fibonacci Quarterly. 1982;20:118-120.
Yalavigi CC. Properties of Tribonacci numbers. 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.
Howard FT, Saidak F. Zhou’s theory of constructing identities. Congress Numer. 2010;200:225-237.
Atassanov K, Dimitriv D, Shannon A. A remark on functions and Pell-Padovan’s Sequence. Notes on Number Theory and Discrete Mathematics. 2009;15(2):1-44.
Bilgici G. Generalized Order–k Pell–Padovan–like numbers by matrix methods. Pure and Applied Mathematics Journal. 2013;2(6):174-178.
DOI:10.11648/j.pamj.20130206.11
Deveci O¨ . The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite Groups. Util. Math. 2015;98:257-270.
Deveci O¨ . Akuzum Y, Karaduman E. “The Pell-Padovan p-sequences and its applications”. Util. Math. 2015;98:327-347.
Deveci O¨ , Shannon AG. Pell–Padovan-Circulant sequences and their applications. Notes on Number Theory and Discrete Mathematics. 2017;23(3):100-114.
TascıD. Padovan and Pell-Padovan quaternions. Journal of Science and Arts. 2018;1(42):125-132.
Tas¸cıD. Acar H. Gaussian Padovan and Gaussian Pell-Padovan numbers. Commun. Fac. Sci. Ank. Ser. A1 Math. Stat. 2018;67(2):82-88.
Tas¸yurdu Y, Akpınar A. Padovan and Pell-Padovan octonions. Proceedings of International Conference on Mathematics and Mathematics Education (ICMME 2019), Turk. J. Math. Comput. Sci. 2019;11(Special Issue):114-122.
Sloane NJA. The on-line encyclopedia of integer sequences.
Available:http://oeis.org/
Kilic¸ E, Stanica P. A matrix approach for general higher order linear Recurrences. Bulletin of the Malaysian Mathematical Sciences Society (2). 2011;34(1):51-67.
Soykan Y. Simson identity of generalized m-step Fibonacci numbers. Int. J. Adv. Appl. Math. and Mech. 2019;7(2):45-56.
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
Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quarterly. 1982;20(1):73-76.
Koshy T. Fibonacci and Lucas numbers with applications. Wiley-Interscience. New York; 2001.
Marohni´c L, Strmeˇcki T. Plastic number: Construction and applications. Advanced Research in Scientific Areas. 2012;3(7):1523-1528.
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Available:https://doi.org/10.1007/978-3-319-00143-2 27
(First published as: Richard Padovan, “Dom Hans van Der Laan and the Plastic Number”, in Nexus IV: Architecture and Mathematics, Kim Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books, 2002;181-193.
Available:http://www.nexusjournal.com/con-ferences/N2002-Padovan.html
Padovan R. Dom Hans van der Laan: Modern primitive. Architectura and Natura Press; 1994.
Catalani M. Identities for Tribonacci-related sequences; 2012. arXiv:math/0209179
Choi E. Modular Tribonacci numbers by matrix method. Journal of the Korean Society of Mathematical Education
Series B: Pure and Applied. Mathematics. 2013;20(3):207-221.
Elia M. Derived sequences, the Tribonacci recurrence and cubic forms. Fibonacci Quarterly. 2001;39(2):107-115.
Er MC. Sums of Fibonacci numbers by matrix methods. Fibonacci Quarterly. 1984;22(3):204-207.
Lin PY. De Moivre-type identities for the Tribonacci numbers. Fibonacci Quarterly. 1988;26:131-134.
Pethe S. Some identities for Tribonacci sequences. Fibonacci Quarterly. 1988;26(2):144-151.
Scott A, Delaney T, Hoggatt Jr. V. The Tribonacci sequence. Fibonacci Quarterly. 1977;15(3):193-200.
Shannon A. Tribonacci numbers and Pascal’s pyramid. Fibonacci Quarterly. 1977;15(3):268&275.
Soykan Y. Tribonacci and Tribonacci-lucas sedenions. Mathematics. 2019;7(1):74.
Spickerman W. Binet’s formula for the Tribonacci sequence. Fibonacci Quarterly. 1982;20:118-120.
Yalavigi CC. Properties of Tribonacci numbers. 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.
Howard FT, Saidak F. Zhou’s theory of constructing identities. Congress Numer. 2010;200:225-237.
Atassanov K, Dimitriv D, Shannon A. A remark on functions and Pell-Padovan’s Sequence. Notes on Number Theory and Discrete Mathematics. 2009;15(2):1-44.
Bilgici G. Generalized Order–k Pell–Padovan–like numbers by matrix methods. Pure and Applied Mathematics Journal. 2013;2(6):174-178.
DOI:10.11648/j.pamj.20130206.11
Deveci O¨ . The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite Groups. Util. Math. 2015;98:257-270.
Deveci O¨ . Akuzum Y, Karaduman E. “The Pell-Padovan p-sequences and its applications”. Util. Math. 2015;98:327-347.
Deveci O¨ , Shannon AG. Pell–Padovan-Circulant sequences and their applications. Notes on Number Theory and Discrete Mathematics. 2017;23(3):100-114.
TascıD. Padovan and Pell-Padovan quaternions. Journal of Science and Arts. 2018;1(42):125-132.
Tas¸cıD. Acar H. Gaussian Padovan and Gaussian Pell-Padovan numbers. Commun. Fac. Sci. Ank. Ser. A1 Math. Stat. 2018;67(2):82-88.
Tas¸yurdu Y, Akpınar A. Padovan and Pell-Padovan octonions. Proceedings of International Conference on Mathematics and Mathematics Education (ICMME 2019), Turk. J. Math. Comput. Sci. 2019;11(Special Issue):114-122.
Sloane NJA. The on-line encyclopedia of integer sequences.
Available:http://oeis.org/
Kilic¸ E, Stanica P. A matrix approach for general higher order linear Recurrences. Bulletin of the Malaysian Mathematical Sciences Society (2). 2011;34(1):51-67.
Soykan Y. Simson identity of generalized m-step Fibonacci numbers. Int. J. Adv. Appl. Math. and Mech. 2019;7(2):45-56.
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
Kalman D. Generalized Fibonacci numbers by matrix methods. Fibonacci Quarterly. 1982;20(1):73-76.
Koshy T. Fibonacci and Lucas numbers with applications. Wiley-Interscience. New York; 2001.
Marohni´c L, Strmeˇcki T. Plastic number: Construction and applications. Advanced Research in Scientific Areas. 2012;3(7):1523-1528.
Padovan R. Dom Hans van Der Laan and the Plastic number. In: Williams K, Ostwald M. (Eds) Architecture and Mathematics from Antiquity to the Future. Birkh¨auser, Cham. 2015;407-419.
Available:https://doi.org/10.1007/978-3-319-00143-2 27
(First published as: Richard Padovan, “Dom Hans van Der Laan and the Plastic Number”, in Nexus IV: Architecture and Mathematics, Kim Williams and Jose Francisco Rodrigues, eds. Fucecchio (Florence): Kim Williams Books, 2002;181-193.
Available:http://www.nexusjournal.com/con-ferences/N2002-Padovan.html
Padovan R. Dom Hans van der Laan: Modern primitive. Architectura and Natura Press; 1994.