C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 67, N umb er 2, Pages 82–88 (2018)
D O I: 10.1501/C om mua1_ 0000000863 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
GAUSSIAN PADOVAN AND GAUSSIAN PELL- PADOVAN SEQUENCES
DURSUN TA¸SCI
Abstract. In this paper, we extend Padovan and Pell- Padovan numbers to Gaussian Padovan and Gaussian Pell-Padovan numbers, respectively. More-over we obtain Binet-like formulas,generating functions and some identities related with Gaussian Padovan numbers and Gaussian Pell-Padovan numbers.
1. Introduction
Horadam [3] in 1963 and Berzsenyi [2] in 1977 de…ned complex Fibonacci num-bers. Horadam introduced the concept the complex Fibonacci numbers as the Gaussian Fibonacci numbers.
Padovan sequence is named after Richard Padovan [7] and Atasonav K., Dimitrov D., Shannon A. and Kritsana S. [1, 4, 5, 6] studied Padovan sequence and Pell-Padovan sequence.
The Padovan sequence is the sequence of integers Pnde…ned by the initial values
P0= P1= P2= 1 and the recurrence relation
Pn= Pn 2+ Pn 3 for all n 3:
The …rst few values of Pn are 1; 1; 1; 2; 2; 3; 4; 5; 7; 9; 12; 16; 21; 28; 37:
Pell-Padovan sequence is de…ned by the initial values R0 = R1 = R2 = 1 and
the recurrence relation
Rn = 2Rn 2+ Rn 3 for all n 3:
The …rst few values of Pell-Padovan numbers are 1; 1; 1; 3; 3; 7; 9; 17; 25; 43; 67; 111; 177; 289:
Received by the editors: January 16, 2017; Accepted: June 12, 2017. 2010 Mathematics Subject Classi…cation. Primary 11B39; Secondary 15B36.
Key words and phrases. Padovan numbers, Pell-Padovan numbers, Gaussian Padovan num-bers, Gaussian Pell-Padovan numbers.
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2. Gaussian Padovan Sequences Firstly we give the de…nition of Gaussian Padovan sequence.
De…nition 2.1. The Gaussian Padovan sequence is the sequence of complex num-bers GPn de…ned by the initial values GP0= 1; GP1= 1 + i; GP2= 1 + i and the
recurrence relation
GPn= GPn 2+ GPn 3 for all n 3:
The …rst few values of GPn are 1; 1 + i; 1 + i; 2 + i; 2 + 2i; 3 + 2i; 4 + 3i; 5 +
4i; 7 + 5i; 9 + 7i:
The following theorem is related with the generating function of the Gaussian Padovan sequence.
Theorem 2.2. The generating function of the Gaussian Padovan sequence is g(x) = 1 + (1 + i) x + i x 2 1 x2 x3 : Proof. Let g(x) = 1 X n=0 GPnxn = GP0+ GP1x + GP2x2+ + GPnxn+
be the generating function of the Gaussian Padovan sequence. On the other hand, since x2g(x) = GP0x2+ GP1x3+ GP2x4+ + GPn 2xn+ and x3g(x) = GP0x3+ GP1x4+ GP2x5+ + GPn 3xn+ we write (1 x2 x3)g(x) = GP0+ GP1x + (GP2 GP0)x2+ (GP3 GP1 GP0)x3 + + (GPn GPn 2 GPn 3)xn+
Now consider GP0 = 1; GP1 = 1 + i; GP2 = 1 + i and GPn = GPn 2+ GPn 3.
Thus, we obtain (1 x2 x3)g(x) = 1 + (1 + i)x + i x2 or g(x) =1 + (1 + i) x + i x 2 1 x2 x3 :
So, the proof is complete.
Now we give Binet-like formula for the Gaussian Padovan sequence. Theorem 2.3. Binet-like formula for the Gaussian Padovan sequence is
GPn= a + i a r1 r1n+ b + ib r2 r2n+ c + ic r3 r3n
where a = (r2 1)(r3 1) (r1 r2)(r1 r3) ; b = (r1 1)(r3 1) (r2 r1)(r2 r3) ; c = (r1 1)(r2 1) (r1 r3)(r2 r3)
and r1; r2; r3 are the roots of the equation x3 x 1 = 0.
Proof. It is easily seen that
GPn = Pn+ iPn 1:
On the other hand, we know that the Binet-like formula for the Padovan sequence is Pn= (r2 1)(r3 1) (r1 r2)(r1 r3) rn1 + (r1 1)(r3 1) (r2 r1)(r2 r3) r2n+ (r1 1)(r2 1) (r1 r3)(r2 r3) r3n: So, the proof is easily seen.
Theorem 2.4.
n
X
j=0
GPj= GPn+ GPn+1+ GPn+2 2(1 + i):
Proof. By the de…nition of Gaussian Padovan sequence recurrence relation GPn= GPn 2+ GPn 3 and GP0 = GP2 GP 1 GP1 = GP3 GP0 GP2 = GP4 GP1 .. . GPn 2 = GPn GPn 3 GPn 1 = GPn+1 GPn 2 GPn = GPn+2 GPn 1 Then we have n X j=0 GPj= GPn+ GPn+1+ GPn+2 GP 1 GP0 GP1:
Now considering GP 1= i; GP0= 1 and GP1= 1 + i, we write n
X
j=0
GPj= GPn+ GPn+1+ GPn+2 2 2i:
Now we investigate the new property of Gaussian Padovan numbers in relation with Padovan matrix formula. We consider the following matrices:
Q3= 2 4 01 10 10 0 1 0 3 5 ; K3= 2 4 1 + i1 + i 1 + i1 1i 1 i 1 3 5 and M3n = 2 4 GPGPn+2n+1 GPGPn+1n GPGPn 1n GPn GPn 1 GPn 2 3 5 : Theorem 2.5. For all n2 Z+,we have
Qn3K3= M3n:
Proof. The proof is easily seen that using the induction on n. Theorem 2.6. If P = 2 4 00 10 01 1 1 0 3 5 then we have 2 4 00 10 01 1 1 0 3 5 n2 4 1 + i1 1 + i 3 5 = 2 4 GPGPn+1n GPn+2 3 5 : Proof. The proof can be seen by mathematical induction on n.
3. Gaussian Pell-Padovan Sequence
As well known Pell-Padovan sequence is de…ned by the recurrence relation Rn= 2Rn 2+ Rn 3; n 3
and initial values are R0= R1= R2= 1.
Now we de…ne Gaussian Pell-Padovan sequence.
De…nition 3.1. The Gaussian Pell-Padovan sequence is de…ned by the recurrence relation
GRn= 2GRn 2+ GRn 3; n 3
and initial values are GR0= 1 i; GR1= 1 + i; GR2= 1 + i:
The …rst few values of GRnare 1 i; 1+i; 1+i; 3+i; 3+3i; 7+3i; 9+7i; 17+9i:
Theorem 3.2. The generating function of Gaussian Pell-Padovan sequence is f (x) = 1 i + (1 + i)x + ( 1 + 3i)x
2
Proof. Let f (x) = 1 X n=0 GRnxn
be the generating function of the Gaussian Pell-Padovan sequence. In this case, we have 2x2f (x) = 2GR0x2+ 2GR1x3+ 2GR2x4+ + 2GRn 2xn+ and x3f (x) = GR0x3+ GR1x4+ GR2x5+ + GRn 3xn+ so we obtain (1 2x2 x3)f (x) = GR0+ GR1x + (GR2 2GR0)x2+ (GR3 2GR1 GR0)x3 + + (GRn 2GRn 2 GRn 3)xn+ :
On the other hand, since GR0 = 1 i; GR1 = 1 + i; GR2 = 1 + i and GRn =
2GRn 2+ GRn 3, then we have
f (x) = 1 i + (1 + i)x + ( 1 + 3i)x
2
1 2x2 x3
which is desired.
Theorem 3.3. The Binet-like formula of Gaussian Pell-Padovan sequence is GRn= 2 p 5 1 + i 1 1 n 2 p 5 1 + i 1 1 n + (i 1) n where =1 + p 5 2 ; = 1 p5 2 ; = 1
are roots of the equation x3 2x 1 = 0:
Proof. The Binet-like formula of Pell-Padovan sequence is given by Rn= 2 n+1 n+1 2 n n + n+1: Now consider GRn= Rn+ iRn 1
so the proof is easily seen.
Theorem 3.4. Pnj=0GRj= 12[( 1 3i) GRn+1+ GRn+2+ GRn+3] : Proof. We …nd that n X j=0 Rj = 1 2( 1 Rn+1+ Rn+2+ Rn+3)
and n X j=0 Rj 1= 1 2( 3 2Rn Rn+1+ Rn+2+ Rn+3): Since GRn= Rn+ iRn 1 we have n X j=0 GRj = n X j=0 Rj+ i n X j=0 Rj 1
So the theorem is proved.
Theorem 3.5. Pnj=1GR2j= R2n+1+ iR2n (n + 1) + i(n 1):
Proof. If we consider the following equalities, then the proof is seen:
n X j=1 R2j = R2n+1 (n + 1) n X j=1 R2j 1 = R2n+ (n 1) and n X j=1 GR2j = n X j=1 R2j+ i n X j=1 R2j 1 Theorem 3.6. Pnj=1 nj GRj= GR2n+ (1 i):
Proof. Considering the following equalities:
n X j=1 n j Rj = R2n+ 1 n X j=1 n j Rj 1 = R2n 1 1 and n X j=1 n j GRj= n X j=1 n j Rj+ i n X j=1 n j Rj 1 then the proof is easily seen.
Now we shall give the new properties of Gaussian Pell-Padovan numbers relation with Pell-Padovan matrix:
Theorem 3.7. If we take the following matrices Q3= 2 4 01 20 10 0 1 0 3 5 ; K3= 2 4 1 + i1 + i 1 + i1 i 11 + 3ii 1 i 1 + 3i 3 5i 3 5 and S3n= 2 4 GRGRn+2n+1 GRGRn+1n GRGRn 1n GRn GRn 1 GRn 2 3 5 : then Qn3:K3= S3n for all n 2 Z+: Theorem 3.8. 2 4 00 10 01 1 2 0 3 5 n2 4 11 + ii 1 + i 3 5 = 2 4 GRGRn+1n GRn+2 3 5 for all n 2 Z+:
We note that for the proofs Theorem 3.7 and Theorem 3.8 are used induction on n.
References
[1] Atassanov, K., Dimitriv, D. and Shannon, A., A remark on functions and Pell-Padovan’s Sequence, Notes on Number Theory and Discrete Mathematics 15(2) (2009), 1–11.
[2] Berzsenyi, Gaussian Fibonacci numbers, The Fibonacci Quarterly,15 (1977) 223-236. [3] Horadam, A.F., Complex Fibonacci Numbers and Fibonacci Quaternions, American
Mathe-matics Monthly 70 (1963) 289-291.
[4] Kritsana, S., Matrices formula for Padovan and Perrin Sequences, Applied Mathematics Sci-ences, 7(142) (2013) 7093-7096.
[5] Shannon, A.G., Anderson, P.G. and Horadam, A.F., Van der Loan numbers, International Journal of Mathematics Education in Science & Technology 37(7) (2006) 825-831.
[6] Shannon, A.G., Anderson, A. F. and Anderson, P.R., The Auxiliary Equation Associated with the Plastic Numbers, Notes on Number Theory and Discrete Mathematics 12(1) (2006) 1-12. [7] Voet, C., The Poetics of order: Dom Hans Van der Loan’s numbers, Architectonic Space,
ARQ.16 (2012) 137-154.
Current address : Gazi University Faculty of Science Department of Mathematics 06500 Teknikokullar-Ankara TURKEY
E-mail address : dtasci@gazi.edu.tr
