Year 2021, Volume 50 , Issue 3, Pages 647 - 658 2021-06-07

An extension of Lucas identity via Pascal's triangle

Giuseppina ANATRIELLO [1] , Giovanni VİNCENZİ [2]


The Fibonacci sequence can be obtained by drawing diagonals in a Pascal’s triangle, and from this, we can obtain the Lucas identity. An investigation on the behavior of certain kinds of other diagonals inside a Pascal’s triangle identifies a new family of recursive sequences: the $k$-Padovan sequences. This family both contains the Fibonacci and the Padovan sequences. A general binomial identity for $k$-Padovan sequences which extends both the well-known Lucas identity and the less known Padovan identity is derived.
combinatorial identities, Pascal, $k$-Fibonacci diagonals, Fibonacci sequence, Padovan sequence
  • [1] Z. Akyuz and S. Halici, On some combinatorial identities involving the terms of generalized Fibonacci and Lucas sequences, Hacet. J. Math. Stat. 42 (4), 431–435, 2013.
  • [2] G. Anatriello and G. Vincenzi, Tribonacci-like sequences and generalized Pascal’s triangles, Internat. J. Math. Ed. Sci. Tech. 45 (8), 1220–1232, 2014.
  • [3] G. Anatriello and G. Vincenzi, Padovan-like sequences and generalized Pascal’s Triangles, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, LXVI (f1), 25–35, 2020.
  • [4] H. Belbachir, T. Komatsu and L. Szalay, Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities, Math. Slovaca 64, 287–300, 2014.
  • [5] J.H. Conway and R.K. Guy, The Book of Numbers, World Scientific, Singapore, 2008.
  • [6] A. Fiorenza and G. Vincenzi, Limit of ratio of consecutive terms for general order-k linear homogeneous recurrences with constant coefficients, Chaos Solitons Fractals 44, 147–152, 2011.
  • [7] N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Math. Comput. Simulation 125, 168–177, 2016.
  • [8] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Solitons Fractals 98, 178–182, 2017.
  • [9] A. Ipek, On (p, q)-Fibonacci quaternions and their Binet’ s Formulas, generating functions and certain binomial sums, Adv. Appl. Clifford Algebras 27 (2), 1343–1351, 2017.
  • [10] G. Kallós , A generalization of Pascal’s triangle using powers of base numbers, Ann. Math. Blaise Pascal 13 (1), 1–15, 2006.
  • [11] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley–Interscience, New York, 2001.
  • [12] L. Maronhič, Plastic Number: Construction and Applications, 5th Virtual International Conference on Advanced Research in Scientific Areas (ARSA-2016) Slovakia, November 7–11, 2016.
  • [13] L. Németh, The trinomial transform triangle, J. Integer Seq. 21 (7), Art. 18.7.3, pp18, 2018.
  • [14] L. Németh and L. Szalay, Recurrence sequences in the hyperbolic Pascal triangle corresponding to the regular mosaic 4,5, Ann. Math. Inform. 46, 165–173, 2016.
  • [15] M.J. Ostwald, Under siege: the golden mean in architecture, Nexus Netw. J. 2, 75–81, 2000.
  • [16] R. Padovan, Proportion: Science, Philosophy, Architecture,1-st Edition, Taylor and Francis, London and New York, 1999.
  • [17] R. Padovan, Dom Hans van der Laan and the Plastic Number, in:Williams K., Ostwald M. (eds) Architecture and Mathematics from Antiquity to the Future, Birkhäuser, 407–419, 2015.
  • [18] P. Szalapaj, Contemporary Architecture and the Digital Design Process, Routledge- Architectural press, New York, 2005.
  • [19] A. Szynal-Liana and I. Wloch, The Fibonacci hybrid numbers, Util. Math. 110, 3–10, 2019.
  • [20] A.G. Shannon, Tribonacci numbers and Pascal’s Pyramid, Fibonacci Quart. 15, 268– 275, 1977.
  • [21] A.G. Shannon, P.G. Anderson and A.F. Horadam, Properties of Cordonnier, Perrin and van der Laan numbers, Int. J. Math. Educ. Sci. Technol. 37 (7), 825–831, 2006.
  • [22] M.Z. Spivey, The Art of Proving Binomial Identities, CRC Press (Taylor and Francis Group), A. Chapman and Hall Book. Boca Raton (FL), 2019.
  • [23] G. Vincenzi and S. Siani, Fibonacci-like sequences and generalized Pascal’s triangles. Int. J. Math. Educ. Sci. Technol. 45 (4), 609–614, 2014.
  • [24] S.-L. Yang, Some identities involving the binomial sequences, Discrete Math. 308 (1), 51–58, 2008.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0003-2168-8169
Author: Giuseppina ANATRIELLO (Primary Author)
Institution: Università di Napoli “Federico II”
Country: Italy


Orcid: 0000-0002-3869-885X
Author: Giovanni VİNCENZİ
Institution: Università di Salerno
Country: Italy


Dates

Publication Date : June 7, 2021

Bibtex @research article { hujms744408, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2021}, volume = {50}, pages = {647 - 658}, doi = {10.15672/hujms.744408}, title = {An extension of Lucas identity via Pascal's triangle}, key = {cite}, author = {Anatrıello, Giuseppina and Vincenzi, Giovanni} }
APA Anatrıello, G , Vincenzi, G . (2021). An extension of Lucas identity via Pascal's triangle . Hacettepe Journal of Mathematics and Statistics , 50 (3) , 647-658 . DOI: 10.15672/hujms.744408
MLA Anatrıello, G , Vincenzi, G . "An extension of Lucas identity via Pascal's triangle" . Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 647-658 <https://dergipark.org.tr/en/pub/hujms/issue/62731/744408>
Chicago Anatrıello, G , Vincenzi, G . "An extension of Lucas identity via Pascal's triangle". Hacettepe Journal of Mathematics and Statistics 50 (2021 ): 647-658
RIS TY - JOUR T1 - An extension of Lucas identity via Pascal's triangle AU - Giuseppina Anatrıello , Giovanni Vincenzi Y1 - 2021 PY - 2021 N1 - doi: 10.15672/hujms.744408 DO - 10.15672/hujms.744408 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 647 EP - 658 VL - 50 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.744408 UR - https://doi.org/10.15672/hujms.744408 Y2 - 2020 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics An extension of Lucas identity via Pascal's triangle %A Giuseppina Anatrıello , Giovanni Vincenzi %T An extension of Lucas identity via Pascal's triangle %D 2021 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 50 %N 3 %R doi: 10.15672/hujms.744408 %U 10.15672/hujms.744408
ISNAD Anatrıello, Giuseppina , Vincenzi, Giovanni . "An extension of Lucas identity via Pascal's triangle". Hacettepe Journal of Mathematics and Statistics 50 / 3 (June 2021): 647-658 . https://doi.org/10.15672/hujms.744408
AMA Anatrıello G , Vincenzi G . An extension of Lucas identity via Pascal's triangle. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 647-658.
Vancouver Anatrıello G , Vincenzi G . An extension of Lucas identity via Pascal's triangle. Hacettepe Journal of Mathematics and Statistics. 2021; 50(3): 647-658.
IEEE G. Anatrıello and G. Vincenzi , "An extension of Lucas identity via Pascal's triangle", Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 3, pp. 647-658, Jun. 2021, doi:10.15672/hujms.744408