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A Study On the Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of Pn k=0 kxkW2

Year 2021, Volume: 5 Issue: 1, 1 - 23, 15.02.2021
https://doi.org/10.26900/jsp.5.1.02

Abstract

In this paper, closed forms of the sum formulas
Pn
k=0 kxkWk
2;
Pn
k=0 kxkWk+2Wk and
Pn
k=0 kxkWk+1Wk
for the squares of generalized Tribonacci numbers are presented. Here, fWmgm2Z is the generalized Tribonacci se-
quence, n is a non-negative integer and x is a real or complex number. As special cases, we give summation formulas
of Tribonacci, Tribonacci-Lucas, Padovan, Perrin numbers and the other third order recurrence relations.
2020 Mathematics Subject Classication. 11B39, 11B83.

References

  • Bruce, I., A modified Tribonacci sequence, Fibonacci Quarterly, 22(3), 244--246, 1984.
  • Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179, 2012.
  • Čerin, Z., Formulae for sums of Jacobsthal--Lucas numbers, Int. Math. Forum, 2(40), 1969--1984, 2007.
  • Čerin, Z., Sums of Squares and Products of Jacobsthal Numbers. Journal of Integer Sequences, 10, Article 07.2.5, 2007.
  • Chen, L., Wang, X., The Power Sums Involving Fibonacci Polynomials and Their Applications, Symmetry, 11, 2019, doi.org/10.3390/sym11050635.
  • Choi, E., Modular Tribonacci Numbers by Matrix Method, Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics. 20(3), 207--221, 2013.
  • Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, Fibonacci Quarterly, 39 (2), 107-115, 2001.
  • Frontczak, R.,Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes on Number Theory and Discrete Mathematics, 24(2), 94--103, 2018.
  • Frontczak, R., Sums of Cubes Over Odd-Index Fibonacci Numbers, Integers, 18, 2018.
  • Gnanam, A., Anitha, B., Sums of Squares Jacobsthal Numbers. IOSR Journal of Mathematics, 11(6), 62-64. 2015.
  • Kiliç, E., Taşçi, D., The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 3(11), 2005.
  • Kılıc, E., Sums of the squares of terms of sequence {u_{n}}, Proc. Indian Acad. Sci. (Math. Sci.) 118(1), 27--41, 2008.
  • Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
  • Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
  • Prodinger, H., Sums of Powers of Fibonacci Polynomials, Proc. Indian Acad. Sci. (Math. Sci.), 119(5), 567-570, 2009.
  • Prodinger, H., Selkirk, S.J., Sums of Squares of Tetranacci Numbers: A Generating Function Approach, 2019, http://arxiv.org/abs/1906.08336v1.
  • Raza, Z., Riaz, M., Ali, M.A., Some Inequalities on the Norms of Special Matrices with Generalized Tribonacci and Generalized Pell-Padovan Sequences, arXiv, 2015, http://arxiv.org/abs/1407.1369v2
  • Schumacher, R., How to sum the squares of the Tetranacci numbers and the Fibonacci m-step numbers. Fibonacci Quarterly, 57:168--175, 2019.
  • Scott, A., Delaney, T., Hoggatt Jr., V., The Tribonacci sequence, Fibonacci Quarterly, 15(3), 193--200, 1977.
  • Shannon, A.G, Horadam, A.F., Some Properties of Third-Order Recurrence Relations, The Fibonacci Quarterly, 10 (2), 135-146, 1972.
  • Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
  • N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
  • Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
  • Soykan, Y., Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports, 9(1), 23-39, 2020. https://doi.org/10.9734/ajarr/2020/v9i130212
  • Soykan Y., Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of and ∑_{k=0}ⁿW_{k}³ and ∑_{k=1}ⁿW_{-k}³, Archives of Current Research International, 20(2), 58-69, 2020. DOI: 10.9734/ACRI/2020/v20i230177
  • Soykan, Y., A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers, Journal of Progressive Research in Mathematics, 16(2), 2932-2941, 2020.
  • Soykan, Y., A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of ∑_{k=0}ⁿx^{k}W_{k}³ and ∑_{k=1}ⁿx^{k}W_{-k}³ , Preprints 2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).
  • Soykan Y., On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of ∑_{k=0}ⁿkW_{k}³ and ∑_{k=1}ⁿkW_{-k}³, Asian Research Journal of Mathematics, 16(6), 37-52, 2020. DOI: 10.9734/ARJOM/2020/v16i630196
  • Soykan, Y., On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of ∑_{k=0}ⁿx^{k}W_{k}², Archives of Current Research International, 20(4), 22-47, 2020. DOI: 10.9734/ACRI/2020/v20i430187
  • Soykan, Y., Formulae For The Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of ∑_{k=0}ⁿkW_{k}², IOSR Journal of Mathematics, 16(4), 1-18, 2020. DOI: 10.9790/5728-1604010118
  • Soykan, Y., A Study on Generalized Fibonacci Numbers: Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}³ and ∑_{k=1}ⁿkx^{k}W_{-k}³ for the Cubes of Terms, Earthline Journal of Mathematical Sciences, 4(2), 297-331, 2020. https://doi.org/10.34198/ejms.4220.297331
  • Soykan Y., Generalized Fibonacci Numbers: Sum Formulas of the Squares of Terms, MathLAB Journal, Vol 5, 46-62, 2020.
  • Soykan, Y., Horadam Numbers: Sum of the Squares of Terms of Sequence, Int. J. Adv. Appl. Math. and Mech. In Presss.
  • Soykan, Y., On Generalized Tetranacci Numbers: Closed Form Formulas of the Sum ∑_{k=0}ⁿW_{k}² of the Squares of Terms, Preprints 2020, 2020050453 (doi: 10.20944/preprints202005.0453.v1).
  • Yalavigi, C.C., A Note on `Another Generalized Fibonacci Sequence', The Mathematics Student. 39, 407--408, 1971.
  • Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quarterly, 10(3), 231--246, 1972.
  • Yilmaz, N., Taskara, N., Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8(39), 1947-1955, 2014.
  • Marcellus E. Waddill, Using Matrix Techniques to Establish Properties of a Generalized Tribonacci Sequence (in Applications of Fibonacci Numbers, Volume 4, G. E. Bergum et al., eds.). Kluwer Academic Publishers. Dordrecht, The Netherlands: pp. 299-308, 1991.
  • Wamiliana., Suharsono., Kristanto, P. E., Counting the sum of cubes for Lucas and Gibonacci Numbers, Science and Technology Indonesia, 4(2), 31-35, 2019.
Year 2021, Volume: 5 Issue: 1, 1 - 23, 15.02.2021
https://doi.org/10.26900/jsp.5.1.02

Abstract

References

  • Bruce, I., A modified Tribonacci sequence, Fibonacci Quarterly, 22(3), 244--246, 1984.
  • Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179, 2012.
  • Čerin, Z., Formulae for sums of Jacobsthal--Lucas numbers, Int. Math. Forum, 2(40), 1969--1984, 2007.
  • Čerin, Z., Sums of Squares and Products of Jacobsthal Numbers. Journal of Integer Sequences, 10, Article 07.2.5, 2007.
  • Chen, L., Wang, X., The Power Sums Involving Fibonacci Polynomials and Their Applications, Symmetry, 11, 2019, doi.org/10.3390/sym11050635.
  • Choi, E., Modular Tribonacci Numbers by Matrix Method, Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics. 20(3), 207--221, 2013.
  • Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, Fibonacci Quarterly, 39 (2), 107-115, 2001.
  • Frontczak, R.,Sums of powers of Fibonacci and Lucas numbers: A new bottom-up approach, Notes on Number Theory and Discrete Mathematics, 24(2), 94--103, 2018.
  • Frontczak, R., Sums of Cubes Over Odd-Index Fibonacci Numbers, Integers, 18, 2018.
  • Gnanam, A., Anitha, B., Sums of Squares Jacobsthal Numbers. IOSR Journal of Mathematics, 11(6), 62-64. 2015.
  • Kiliç, E., Taşçi, D., The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 3(11), 2005.
  • Kılıc, E., Sums of the squares of terms of sequence {u_{n}}, Proc. Indian Acad. Sci. (Math. Sci.) 118(1), 27--41, 2008.
  • Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
  • Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
  • Prodinger, H., Sums of Powers of Fibonacci Polynomials, Proc. Indian Acad. Sci. (Math. Sci.), 119(5), 567-570, 2009.
  • Prodinger, H., Selkirk, S.J., Sums of Squares of Tetranacci Numbers: A Generating Function Approach, 2019, http://arxiv.org/abs/1906.08336v1.
  • Raza, Z., Riaz, M., Ali, M.A., Some Inequalities on the Norms of Special Matrices with Generalized Tribonacci and Generalized Pell-Padovan Sequences, arXiv, 2015, http://arxiv.org/abs/1407.1369v2
  • Schumacher, R., How to sum the squares of the Tetranacci numbers and the Fibonacci m-step numbers. Fibonacci Quarterly, 57:168--175, 2019.
  • Scott, A., Delaney, T., Hoggatt Jr., V., The Tribonacci sequence, Fibonacci Quarterly, 15(3), 193--200, 1977.
  • Shannon, A.G, Horadam, A.F., Some Properties of Third-Order Recurrence Relations, The Fibonacci Quarterly, 10 (2), 135-146, 1972.
  • Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
  • N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
  • Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
  • Soykan, Y., Closed Formulas for the Sums of Squares of Generalized Fibonacci Numbers, Asian Journal of Advanced Research and Reports, 9(1), 23-39, 2020. https://doi.org/10.9734/ajarr/2020/v9i130212
  • Soykan Y., Closed Formulas for the Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of and ∑_{k=0}ⁿW_{k}³ and ∑_{k=1}ⁿW_{-k}³, Archives of Current Research International, 20(2), 58-69, 2020. DOI: 10.9734/ACRI/2020/v20i230177
  • Soykan, Y., A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers, Journal of Progressive Research in Mathematics, 16(2), 2932-2941, 2020.
  • Soykan, Y., A Study On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of ∑_{k=0}ⁿx^{k}W_{k}³ and ∑_{k=1}ⁿx^{k}W_{-k}³ , Preprints 2020, 2020040437 (doi: 10.20944/preprints202004.0437.v1).
  • Soykan Y., On Sums of Cubes of Generalized Fibonacci Numbers: Closed Formulas of ∑_{k=0}ⁿkW_{k}³ and ∑_{k=1}ⁿkW_{-k}³, Asian Research Journal of Mathematics, 16(6), 37-52, 2020. DOI: 10.9734/ARJOM/2020/v16i630196
  • Soykan, Y., On the Sums of Squares of Generalized Tribonacci Numbers: Closed Formulas of ∑_{k=0}ⁿx^{k}W_{k}², Archives of Current Research International, 20(4), 22-47, 2020. DOI: 10.9734/ACRI/2020/v20i430187
  • Soykan, Y., Formulae For The Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of ∑_{k=0}ⁿkW_{k}², IOSR Journal of Mathematics, 16(4), 1-18, 2020. DOI: 10.9790/5728-1604010118
  • Soykan, Y., A Study on Generalized Fibonacci Numbers: Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}³ and ∑_{k=1}ⁿkx^{k}W_{-k}³ for the Cubes of Terms, Earthline Journal of Mathematical Sciences, 4(2), 297-331, 2020. https://doi.org/10.34198/ejms.4220.297331
  • Soykan Y., Generalized Fibonacci Numbers: Sum Formulas of the Squares of Terms, MathLAB Journal, Vol 5, 46-62, 2020.
  • Soykan, Y., Horadam Numbers: Sum of the Squares of Terms of Sequence, Int. J. Adv. Appl. Math. and Mech. In Presss.
  • Soykan, Y., On Generalized Tetranacci Numbers: Closed Form Formulas of the Sum ∑_{k=0}ⁿW_{k}² of the Squares of Terms, Preprints 2020, 2020050453 (doi: 10.20944/preprints202005.0453.v1).
  • Yalavigi, C.C., A Note on `Another Generalized Fibonacci Sequence', The Mathematics Student. 39, 407--408, 1971.
  • Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quarterly, 10(3), 231--246, 1972.
  • Yilmaz, N., Taskara, N., Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8(39), 1947-1955, 2014.
  • Marcellus E. Waddill, Using Matrix Techniques to Establish Properties of a Generalized Tribonacci Sequence (in Applications of Fibonacci Numbers, Volume 4, G. E. Bergum et al., eds.). Kluwer Academic Publishers. Dordrecht, The Netherlands: pp. 299-308, 1991.
  • Wamiliana., Suharsono., Kristanto, P. E., Counting the sum of cubes for Lucas and Gibonacci Numbers, Science and Technology Indonesia, 4(2), 31-35, 2019.
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Basic Sciences and Engineering
Authors

Yüksel Soykan This is me 0000-0002-1895-211X

Publication Date February 15, 2021
Published in Issue Year 2021 Volume: 5 Issue: 1

Cite

APA Soykan, Y. (2021). A Study On the Sums of Squares of Generalized Tribonacci Numbers: Closed Form Formulas of Pn k=0 kxkW2. Journal of Scientific Perspectives, 5(1), 1-23. https://doi.org/10.26900/jsp.5.1.02