A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers
Keywords:
Sum of squares, third order recurrence, Tribonacci numbers, Padovan numbers, Perrin numbers, Narayana numbers.
Abstract
In this paper, closed forms of the sum formulas for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of the squares of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences.
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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.
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.
Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
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.
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 2019.
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.
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.
Wamiliana., Suharsono., Kristanto, P. E., Counting the sum of cubes for Lucas and Gibonacci Numbers, Science and Technology Indonesia, 4(2), 31-35, 2019.
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.
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.
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.
Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
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.
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 2019.
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.
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.
Wamiliana., Suharsono., Kristanto, P. E., Counting the sum of cubes for Lucas and Gibonacci Numbers, Science and Technology Indonesia, 4(2), 31-35, 2019.
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.
Published
2020-05-14
How to Cite
Soykan, Y. (2020). A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers. Journal of Progressive Research in Mathematics, 16(2), 2932-2941. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/1845
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