Square numbers, square pyramidal numbers, and generalized Fibonacci polynomials

In this paper, we derive two interesting formulas for square and square pyramidal numbers. We focus on the linear recurrence relation with constant coefficients for square and square pyramidal numbers. Then we deal with the relationship between generalized Fibonacci polynomials and these numbers. Also, we give some determinant representations of these numbers.

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  • OEIS, The on-line encyclopedia of integer sequences, Retrieved April 15, 2022, https://oeis.org/A000290.
  • OEIS, The on-line encyclopedia of integer sequences, Retrieved April 15, 2022, https://oeis.org/A000330.
  • C. Alsina, R. B. Nelsen, A mathematical space odyssey: Solid geometry in the 21st century (the dolciani mathematical expositions), Mathematical Association of America, Washington, DC, 2015.
  • J. H. Conway, R. Guy, The book of numbers, Springer, 1998.
  • R. Grassl, 79.33 the squares do fit!, TheMathematical Gazette, 79(485), (1995) 361–364.
  • P. J. Federico, Descartes on polyhedra: A study of the de solidorum elementis, sources in the history of mathematics and physical sciences, Springer, 1982.
  • T. L. Heath, The works of archimedes, Cambridge University Press, 1897.
  • L. E. Sigler, Fibonacci’s liber abaci, Springer, New York, 2002.
  • W. S. Anglin, The square pyramid puzzle, The American Mathematical Monthly, 97(2), (1990) 120–124.
  • M. Parker, Things to make and do in the fourth dimension: A mathematician’s journey through narcissistic numbers, optimal dating algorithms, at least two kinds of infinity, and more, Farrar, Straus and Giroux, New York, 2015.
  • B. Babcock, A. V. Tuyl, Revisiting the spreading and covering numbers, The Australasian Journal of Combinatorics, 56, (2013) 77–84.
  • R. P. Agarwal, Pythagoreans figurative numbers: The beginning of number theory and summation of series, Journal of Applied Mathematics and Physics, 9, (2021) 2038–2113.
  • OEIS, The on-line encyclopedia of integer sequences, Retrieved April 15, 2022, https://oeis.org/wiki/Index_to_OEIS: Section_Rec.
  • T. Machenry, A subgroup of the group of units in the ring of arithmetic functions, RockyMountain Journal of Mathematics, 29, (1999) 1055–1065.
  • T. Machenry, K. Wong, Degree k linear recursions mod (pfields, RockyMountain Journal of Mathematics, 41(4), (2011) 1303–1327.
  • H. Li, T. Machenry, Permanents and determinants, weighted isobaric polynomials, and integer sequences, Journal of Integer Sequences, 16, (2013) Article 13.3.5.
  • T. Machenry, Generalized fibonacci and lucas polynomials and multiplicative arithmetic functions, Fibonacci Quarterly, 38(2), (2000) 167–173.
  • T. Machenry, G. Tudose, Reflections on symmetric polynomials and arithemetic functions, Rocky Mountain Journal ofMathematics, 35(3), (2006) 901–928.
  • N. D. Cahill, J. R. D’Errico, D. A. Narayan, J. Y. Narayan, Fibonacci determinants, The College Mathematics Journal, 33, (2002) 221–225.
Journal of New Results in Science-Cover
  • ISSN: 1304-7981
  • Yayın Aralığı: Yılda 3 Sayı
  • Başlangıç: 2012
  • Yayıncı: Tokat Gaziosmanpaşa Üniversitesi