Combinatorial Interpretation of Generalized Pell Numbers

In this note we give combinatorial interpretations for the generalized Pell sequence of order k by means of lattice paths and generalized bi-colored compositions. We also derive some basic relations and identities by using Riordan arrays.

[1]  Thomas Koshy,et al.  Fibonacci and Lucas Numbers with Applications: Koshy/Fibonacci , 2001 .

[2]  Renzo Sprugnoli,et al.  Riordan arrays and combinatorial sums , 1994, Discret. Math..

[3]  T. He,et al.  A unified approach for the Catalan matrices by using Riordan arrays , 2018, Linear Algebra and its Applications.

[4]  Emrah Kilic,et al.  On The Usual Fibonacci and Generalized Order-k Pell Numbers , 2008, Ars Comb..

[5]  Sheng-Liang Yang,et al.  Some matrix identities on colored Motzkin paths , 2017, Discret. Math..

[6]  J. L. Ramírez,et al.  Generalized Schröder matrix and its combinatorial interpretation , 2018 .

[7]  V. Sirvent,et al.  A $q$-analogue of the Biperiodic Fibonacci Sequence , 2015, 1501.05830.

[8]  Gregory P. Dresden,et al.  A Simplified Binet Formula for k-Generalized Fibonacci Numbers , 2009, J. Integer Seq..

[9]  J. Muskat.  Generalized Fibonacci and Lucas sequences and rootfinding methods , 1993 .

[10]  W. Marsden.  I and J , 2012 .

[11]  Richard P. Brent,et al.  On the periods of generalized Fibonacci recurrences , 1994, ArXiv.

[12]  José L. Ramírez,et al.  A Generalization of the $k$-Bonacci Sequence from Riordan Arrays , 2015, Electron. J. Comb..

[13]  A. Benjamin,et al.  Proofs that Really Count: The Art of Combinatorial Proof , 2003 .

[14]  Douglas G. Rogers,et al.  Underdiagonal Lattice Paths with Unrestricted Steps , 1999, Discret. Appl. Math..

[15]  Louis W. Shapiro,et al.  The Riordan group , 1991, Discret. Appl. Math..

[16]  Sheng-liang Yang,et al.  Half of a Riordan array and restricted lattice paths , 2018 .

[17]  Thomas Koshy,et al.  Pell and Pell–Lucas Numbers with Applications , 2014 .