Asian Journal of Advanced Research and Reports 11(2): 8-28, 2020; Article no.AJARR.57839 ISSN: 2582-3248 Generalized Pell-Padovan Numbers ∗ Y ¨ukselSoykan1 1Department of Mathematics, Faculty of Art and Science, Zonguldak Bulent¨ Ecevit University, 67100, Zonguldak, Turkey. Author’s contribution The sole author designed, analyzed, interpreted and prepared the manuscript. Article Information DOI: 10.9734/AJARR/2020/v11i230259 Editor(s): (1) Dr. Hasan Aydogan , Selcuk University, Turkey. Reviewers: (1) B. Selvaraj, Nehru Institute of Engineering and Technology, India. (2) Md. Asaduzzaman, Islamic University, Bangladesh. Complete Peer review History: http://www.sdiarticle4.com/review-history/57839 Received 05 April 2020 Accepted 12 June 2020 Original Research Article Published 20 June 2020 ABSTRACT In this paper, we investigate the generalized Pell-Padovan sequences and we deal with, in detail, four special cases, namely, Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Keywords: Pell-Padovan numbers; Pell-Perrin numbers; third order Fibonacci-Pell numbers; third order Lucas-Pell numbers. 2010 Mathematics Subject Classification: 11B39, 11B83. 1 INTRODUCTION and third order Lucas-Pell. Before, we recall the generalized Tribonacci sequence and its some The aim of this paper is to define and to explore properties. some of the properties of generalized Pell- Padovan numbers and is to investigate, in details, The generalized Tribonacci sequence four particular case, namely sequences of Pell- fWn(W0;W1;W2; r; s; t)gn≥0 (or shortly Padovan, Pell-Perrin, third order Fibonacci-Pell fWngn≥0) is defined as follows: *Corresponding author: E-mail: yuksel [email protected]; Soykan; AJARR, 11(2): 8-28, 2020; Article no.AJARR.57839 Wn = rWn−1 + sWn−2 + tWn−3;W0 = a; W1 = b; W2 = c; n ≥ 3 (1.1) where W0;W1;W2 are arbitrary complex (or real) numbers and r; s; t are real numbers. This sequence has been studied by many authors, see for example [1,2,3,4,5,6,7,8,9,10,11,12,13]. The sequence fWngn≥0 can be extended to negative subscripts by defining s r 1 W− = − W− − − W− − + W− − n t (n 1) t (n 2) t (n 3) for n = 1; 2; 3; ::: when t =6 0: Therefore, recurrence (1.1) holds for all integer n: As fWng is a third order recurrence sequence (difference equation), it’s characteristic equation is x3 − rx2 − sx − t = 0 (1.2) whose roots are r α = α(r; s; t) = + A + B 3 r β = β(r; s; t) = + !A + !2B 3 r γ = γ(r; s; t) = + !2A + !B 3 where ( ) ( ) r3 rs t p 1=3 r3 rs t p 1=3 A = + + + ∆ ;B = + + − ∆ 27 6 2 27 6 2 p r3t r2s2 rst s3 t2 −1 + i 3 ∆ = ∆(r; s; t) = − + − + ;! = = exp(2πi=3) 27 108 6 27 4 2 Note that we have the following identities α + β + γ = r; αβ + αγ + βγ = −s; αβγ = t: If ∆(r; s; t) > 0; then the Equ. (1.2) has one real (α) and two non-real solutions with the latter being conjugate complex (in our case all roots are reals). So, in this case, it is well known that generalized Tribonacci numbers can be expressed, for all integers n; using Binet’s formula b αn b βn b γn W = 1 + 2 + 3 (1.3) n (α − β)(α − γ) (β − α)(β − γ) (γ − α)(γ − β) where b1 = W2 − (β + γ)W1 + βγW0; b2 = W2 − (α + γ)W1 + αγW0; b3 = W2 − (α + β)W1 + αβW0: Note that the Binet form of a sequence satisfying (1.2) for non-negative integers is valid for all integers n; for a proof of this result see [14]: This result of Howard and Saidak [14] is even true in the case of higher-order recurrence relations. In this paper we consider the case r = 0; s = 2; t = 1 and in this case we write Vn = Wn: A generalized Pell-Padovan sequence fVngn≥0 = fVn(V0;V1;V2)gn≥0 is defined by the third-order recurrence relations Vn = 2Vn−2 + Vn−3 (1.4) 9 Soykan; AJARR, 11(2): 8-28, 2020; Article no.AJARR.57839 with the initial values V0 = c0;V1 = c1;V2 = c2 not all being zero. The sequence fVngn≥0 can be extended to negative subscripts by defining V−n = −2V−(n−1) + V−(n−3) for n = 1; 2; 3; :::: Therefore, recurrence (1.4) holds for all integer n: (1.3) can be used to obtain Binet formula of generalized Pell-Padovan numbers. Binet formula of generalized padovan numbers can be given as b αn b βn b γn V = 1 + 2 + 3 n (α − β)(α − γ) (β − α)(β − γ) (γ − α)(γ − β) where b1 = V2 − (β + γ)V1 + βγV0; b2 = V2 − (α + γ)V1 + αγV0; b3 = V2 − (α + β)V1 + αβV0: (1.5) Here, α, β and γ are the roots of the cubic equation x3 − 2x − 1 = 0: Moreover p 1 + 5 α = ; 2p 1 − 5 β = ; 2 γ = −1: Note that α + β + γ = 0; αβ + αγ + βγ = −2; αβγ = 1: The first few generalized Pell-Padovan numbers with positive subscript and negative subscript are given in the following Table 1. Table 1. A few generalized Pell-Padovan numbers n Vn V−n 0 V0 ::: 1 V1 V2 − 2V0 2 V2 −2V2 + V1 + 4V0 3 2V1 + V0 4V2 − 2V1 − 7V0 4 2V2 + V1 −7V2 + 4V1 + 12V0 5 V2 + 4V1 + 2V0 12V2 − 7V1 − 20V0 6 4V2 + 4V1 + V0 −20V2 + 12V1 + 33V0 7 4V2 + 9V1 + 4V0 33V2 − 20V1 − 54V0 8 9V2 + 12V1 + 4V0 −54V2 + 33V1 + 88V0 9 12V2 + 22V1 + 9V0 88V2 − 54V1 − 143V0 10 1022V2 + 33V1 + 12V0 −143V2 + 88V1 + 232V0 Now we define four special cases of the sequence fVng. Pell-Padovan sequencefRngn≥0, Pell- Perrin sequence fCngn≥0; third order Fibonacci-Pell sequence fGngn≥0 and third order Lucas-Pell 10 Soykan; AJARR, 11(2): 8-28, 2020; Article no.AJARR.57839 sequence fBngn≥0 are defined, respectively, by the third-order recurrence relations Rn+3 = Rn+1 + Rn;R0 = 1;R1 = 1;R2 = 1; Cn+3 = Cn+1 + Cn;C0 = 3;C1 = 0;C2 = 2; Gn+3 = Gn+1 + Gn;G0 = 1;G1 = 0;G2 = 2; Bn+3 = Bn+1 + Bn;B0 = 3;B1 = 0;B2 = 4: The sequences fRngn≥0; fCngn≥0; fGngn≥0 and fBngn≥0 can be extended to negative subscripts by defining R−n = −2R−(n−1) + R−(n−3) (1.6) C−n = −2C−(n−1) + C−(n−3) (1.7) G−n = −2G−(n−1) + G−(n−3) (1.8) B−n = −2B−(n−1) + B−(n−3) (1.9) for n = 1; 2; 3; ::: respectively. Therefore, recurrences (1.6), (1.7), (1.8) and (1.9) hold for all integer n: For more information on Pell-Padovan sequence, see [15,16,17,18,19,20,21,22]. Next, we present the first few values of the Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell numbers with positive and negative subscripts: Table 2. The first few values of the special third-order numbers with positive and negative subscripts n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Rn 1 1 1 3 3 7 9 17 25 43 67 111 177 289 R−n −1 3 −5 9 −15 25 −41 67 −109 177 −287 465 −753 Cn 3 0 2 3 4 8 11 20 30 51 80 132 211 344 C−n −4 8 −13 22 −36 59 −96 156 −253 410 −664 1075 −1740 Gn 1 0 2 1 4 4 9 12 22 33 56 88 145 232 G−n 0 0 1 −2 4 −7 12 −20 33 −54 88 −143 232 Bn 3 0 4 3 8 10 19 28 48 75 124 198 323 520 B−n −2 4 −5 8 −12 19 −30 48 −77 124 −200 323 −522 For all integers n; Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell numbers (using initial conditions in (1.5)) can be expressed using Binet’s formulas as 1 n 1 n n Rn = (1 − p )α + (1 + p )β − γ ; 5 5 3 n 3 n n Cn = (2 − p )α + (2 + p )β − γ ; 5 5 1 n 1 n n Gn = p α − p β + γ ; 5 5 n n n Bn = α + β + γ ; respectively. Rn is the sequence A066983 in [23] associated with the relation n Rn+2 = Rn+1 + Rn + (−1) ; with R1 = R2 = 1: Cn is not indexed in [23]. 11 Soykan; AJARR, 11(2): 8-28, 2020; Article no.AJARR.57839 Gn is the sequence A008346 in [23] associated with the relation n Gn = Fn + (−1) where Fn is Fibonacci sequence which is given as Fn = Fn−1 + Fn−2 with F0 = 0 and F1 = 1: Bn is the sequence A099925 in [23] associated with the relation n Bn = Ln + (−1) where Ln is Lucas sequence which is given as Ln = Ln−1 + Ln−2 with L0 = 2 and L1 = 1: Since n+1 n F−n = (−1) Fn and L−n = (−1) Ln we get n+1 n n G−n = (−1) Gn + 1 + (−1) = (−1) (1 − Fn) and n n n B−n = (−1) Bn − 1 + (−1) = (−1) (Ln + 1): 2 GENERATING FUNCTIONS P1 n Next, we give the ordinary generating function Vnx of the sequence Vn: n=0 P1 n Lemma 2.1.