An extension of Lucas identity via Pascal’s triangle. (English) Zbl 1488.05020
Summary: 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.
MSC:
| 05A19 | Combinatorial identities, bijective combinatorics |
| 65Q30 | Numerical aspects of recurrence relations |
| 11Y55 | Calculation of integer sequences |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
Keywords:
combinatorial identities; Pascal triangle; \(k\)-Fibonacci diagonals; Fibonacci sequence; Padovan sequenceReferences:
| [1] | [13] L. Németh, The trinomial transform triangle, J. Integer Seq. 21 (7), Art. 18.7.3, pp18, 2018. · Zbl 1439.11041 |
| [2] | [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. · Zbl 1374.11019 |
| [3] | [15] M.J. Ostwald, Under siege: the golden mean in architecture, Nexus Netw. J. 2, 75-81, 2000. · Zbl 1065.01520 |
| [4] | [16] R. Padovan, Proportion: Science, Philosophy, Architecture,1-st Edition, Taylor and Francis, London and New York, 1999. |
| [5] | [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. |
| [6] | [18] P. Szalapaj, Contemporary Architecture and the Digital Design Process, Routledge- Architectural press, New York, 2005. |
| [7] | [19] A. Szynal-Liana and I. Wloch, The Fibonacci hybrid numbers, Util. Math. 110, 3-10, 2019. · Zbl 1433.11015 |
| [8] | [20] A.G. Shannon, Tribonacci numbers and Pascal’s Pyramid, Fibonacci Quart. 15, 268- 275, 1977. · Zbl 0385.05006 |
| [9] | [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. · Zbl 1149.11300 |
| [10] | [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. · Zbl 1419.05002 |
| [11] | [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. · Zbl 1296.97006 |
| [12] | [24] S.-L. Yang, Some identities involving the binomial sequences, Discrete Math. 308 (1), 51-58, 2008. · Zbl 1127.05006 |
| [13] | [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. · Zbl 1338.11014 |
| [14] | [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. · Zbl 1316.97001 |
| [15] | [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. · Zbl 1474.05017 |
| [16] | [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. · Zbl 1349.11037 |
| [17] | [5] J.H. Conway and R.K. Guy, The Book of Numbers, World Scientific, Singapore, 2008. · Zbl 0866.00001 |
| [18] | [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. · Zbl 1258.11015 |
| [19] | [7] N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Math. Comput. Simulation 125, 168-177, 2016. · Zbl 1527.11012 |
| [20] | [8] S. Halici and A. Karataş, On a generalization for Fibonacci quaternions, Chaos Solitons Fractals 98, 178-182, 2017. · Zbl 1372.11015 |
| [21] | [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. · Zbl 1420.11036 |
| [22] | [10] G. Kallós , A generalization of Pascal’s triangle using powers of base numbers, Ann. Math. Blaise Pascal 13 (1), 1-15, 2006. · Zbl 1172.11302 |
| [23] | [11] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001. · Zbl 0984.11010 |
| [24] | [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. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
