On the Third-Order Horadam and Geometric Mean Sequences
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DOI: 10.1007/s13226-020-0454-0
ON THE THIRD-ORDER HORADAM AND GEOMETRIC MEAN SEQUENCES Gamaliel Cerda-Morales Instituto de Matem´aticas Pontificia Universidad Cat´olica de Valpara´ıso, Blanco Viel 596, Valpara´ıso, Chile e-mail: [email protected] (Received 2 January 2019; after final revision 29 May 2019; accepted 14 June 2019) In this paper, in considering aspects of the geometric mean sequence, offers new results connecting generalized Tribonacci and third-order Horadam numbers which are established and then proved independently. Key words : Generalized Fibonacci number; generalized Tribonacci number; geometric mean sequence; third-order Horadam number. 2010 Mathematics Subject Classification : 11B39, 11K31.
1. I NTRODUCTION The Horadam numbers have many interesting properties and applications in many fields of science (see, e.g., [7, 8]). The Horadam numbers {wn (a, b; r, s)} or {wn } are defined by the recurrence relation w0 = a, w1 = b, wn+2 = rwn+1 + swn ,
(1.1)
where a, b, r and s are real numbers. In [6, 12] the Horadam recurrence relation (1.1) is extended to higher order recurrence relations and the basic list of identities provided by Horadam is expanded and extended to several identities for some of the higher order cases. Furthermore, third-order Horadam numbers, {Hn (a, b, c; r, s, t)} is defined by Hn+3 = rHn+2 + sHn+1 + tHn , H0 = a, H1 = b, H2 = c, n ≥ 0. where a, b, c, r, s and t are arbitrary real numbers.
(1.2)
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GAMALIEL CERDA-MORALES
Some of the following properties given for third-order Horadam numbers are revisited in this paper (for more details, see [1, 4, 5, 6, 12]). Hn+m = hn Hm+1 + (shn−1 + thn−2 ) Hm + thn−1 Hm−1 ,
(1.3)
h2n + sh2n−1 + 2thn−1 hn−2 = h2n−1
(1.4)
and
( Hn2
+
2 sHn−1
+ 2tHn−1 Hn−2 =
cH2n−2 + (sb + ta) H2n−3 +tbH2n−4 ,
) (1.5)
,
where n ≥ 2 and m ≥ 1. Here, the sequence {hn } is the particular case {Hn (0, 1, r; r, s, t)}. As the elements of this Tribonacci-type number sequence provide third order iterative relation, its characteristic equation is x3 − rx2 − sx − t = 0, whose roots are α = and ω2 =
r 3
r 3
+ A + B, ω1 =
r 3
+ ²A + ²2 B
+ ²2 A + ²B, where r A=
3
r B= with ∆ = ∆(r, s, t) =
r3 t 27
−
r 2 s2 108
+
rst 6
3
−
rs t √ r3 + + + ∆, 27 6 2 rs t √ r3 + + − ∆, 27 6 2
s3 27
+
t2 4
and ² = − 21 +
√ i 3 2 .
In this paper, ∆ > 0, then the cubic equation x3 − rx2 − sx − t = 0 has one real and two nonreal solutions, the latter being conjugate complex. Thus, the Binet formula for the third-order Horadam numbers can be expressed as: Hn =
Qω1n Rω2n P αn − + , (α − ω1 )(α − ω2 ) (α − ω1 )(ω1 − ω2 ) (α − ω2 )(ω1 − ω2 )
(1.6)
where the coefficients are P = c − (ω1 + ω2 )b + ω1 ω2 a, Q = c − (α + ω2 )b + αω2 a and R = c − (α + ω1 )b + αω1 a. In particular, if a = 0, b = 1 and c = r, we obtain P = α, Q = ω1 and R = ω2 in Eq. (1.6). Furthermore, the third-order Horadam sequence is the generalization of the well-known sequences Tribonacci, Padovan, Narayana and third-order Jacobsthal (see [3, 5]). Let a, b and c be real numbers. Cons
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