Research Article Gaussian Fibonacci Circulant Type Matrices

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1 Hidawi Publishig Corporatio Abstract ad Applied Aalysis Article ID pages Research Article Gaussia Fiboacci Circulat Type Matrices Zhaoli Jiag Hogxia Xi 2 ad Fuliag Lu Departmet of Mathematics Liyi Uiversity Liyi Shadog Chia 2 Departmet of Mathematics Shadog Normal Uiversity Ji a Shadog Chia Correspodece should be addressed to Hogxia Xi; xhogx052@63com Received 4 July 204; Accepted 4 August 204; Published 3 August 204 Academic Editor: Huahe Dog Copyright 204 Zhaoli Jiag et al This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited Circulat matrices have become importat tools i solvig itegrable system Hamiltoia structure ad itegral equatios I this paper we prove that Gaussia Fiboacci circulat type matrices are ivertible matrices for >2ad give the explicit determiats ad the iverse matrices Furthermore the upper bouds for the spread o Gaussia Fiboacci circulat ad left circulat matrices are preseted respectively Itroductio Circulat matrices have bee used i solvig itegrable system [] Hamiltoia structure [2 3] ad itegral equatios [4 8] By usig the KdV ad Boussiesq systems the circulat forward shift matrix ad the atisymmetric circulat matrix Weiss i [] costructed a cosymplectic form M ξ ad presets the factorizatio of the BLP equatio by the periodic fixed poits of its Bäcklud trasformatios I [2] Kupershmidt ad Wilso by Propositio 8 [2] showedthat a first Hamiltoia structure for their modified equatios formed from circulat operators P [2] does ot exist sice the relevat Hamiltoias H LP [2] do ot survive the specializatio However the Hamiltoias H P [2] dosurvive;the they verified that a secod Hamiltoia structure exists I [3] Kisisel solved the problem o the Hamiltoia structure of discrete KP equatios by the properties of circulat matrices Cha et al cosidered solvig potetial equatios by the boudary itegral equatio approach The equatios derived are Fredholm itegral equatios of the first kid ad are kow to be ill-coditioed They proposed to solve the equatios by the precoditioed cojugate gradiet method with circulat itegral operators as precoditioers i [4] By miimizig the problem W m A m F [5] obtaiedfrom the precoditioers of type block circulat with circulat blocks ad i the case that the coefficiet matrix of C A is positive defiite Malekejad ad Rabbai used CG method for solvig system of the C AxC b i [5] Gohberg et al stated that fiite sectios of a Wieer-Hopf itegral operator ca be approximated by circulat itegral operators withi a sum of a small ad a fiite rak operator i [6] They gave two costructios of such circulat operators which ca be used to accelerate covergece of the CG algorithm as applied to fiite sectios of a Wieer-Hopf equatio Cai developed a fast ad direct Fourier spectral method for solvig the Hilbert type sigular itegral equatio Whe the direct Fourier spectral method is used to solve ( i [7] Cai observed that the matrix represetatio of operator A uder the Fourier basis is a quasicirculat matrix Abramya proved the solvability of the system of (5 [8]forall Nbegiig with some 0 viathataycirculatmatrixisaormalmatrix ad its spectral orm is equal to the maximal modulus of its eigevalues Circulat type matrices have bee put o the firm basis with the work i [9 3]adsooTherearediscussiosabout the covergece i probability ad i distributio of the spectral orm of circulat type matrices i [4] Furthermore the g-circulat matrices are focused o by may researchers; for more details please refer to [5 7] ad the refereces therei Recetly some authors gave the explicit determiat ad iverse of the circulat ad skew-circulat ivolvig famous umbers Cambii preseted a explicit form of the iverse of a particular circulat matrix i [8] Jiag et al [9] cosidered circulat type matrices with the k-fiboacci ad k-lucas umbers ad preseted the explicit determiat ad

2 2 Abstract ad Applied Aalysis iverse matrix by costructig the trasformatio matrices I [20] Jiag ad Hog preseted exact determiats of some special circulat matrices ivolvig four kids of famous umbers Bozkurt ad Tam gave determiats ad iverses of circulat matrices with Jacobsthal ad Jacobsthal-Lucas umbers i [2] I [22] authors studied the osigularity of the skew circulat type matrices ad preseted explicit determiats ad iverse matrices of these special matrices Furthermore four kids of orms ad bouds for the spread of these matrices are give separately She et al cosidered circulat matrices with Fiboacci ad Lucas umbers ad preseted their explicit determiats ad iverses i [23] Jiag ad Li [24] discussed the osigularity of the circulat type matrix ad gave the explicit determiat ad iverse matrices The Gaussia Fiboacci sequece [25 26] is defied by the followig recurrece relatios: G G G ( with the iitial coditio G 0 i G G F if F is the th Fiboacci umber i The {G } is give by the formula G ( iβ α (iα β α β (α β i α β α β (2 α ad β are the roots of the characteristic equatio x 2 x 0 I this paper circulat type matrices iclude the circulat left circulat ad g-circulat matrices Let r be a oegative iteger We defie a Gaussia Fiboacci circulat matrix which is a complex matrix with the followig form: G r2 G r G r G r Circ ( G r2 G r [ ] [ G r2 G r3 ] (3 Besides a Gaussia Fiboacci left circulat matrix is give by G r2 G r G r2 G r3 LCirc ( G r2 G r [ ] [ G r G r ] (4 each row is a cyclic shift of the row above to the left A Gaussia Fiboacci g-circulat matrix is a complex matrix with the followig form: G r2 G r G r g G r g2 G r g A gr ( G r 2g G r 2g2 G r 2g (5 d G rg G rg2 G rg g is a oegative iteger ad each of the subscripts is uderstoodtobereducedmodulo The first row of A g is ( G r2 G r ad its (j throwisobtaiedbygivigitsjth row a right circular shift by g positios (equivaletly g mod positios Note that gor gyields the Gaussia Fiboacci circulat matrix If g the we obtai the Gaussia Fiboacci left circulat matrix 2 Determiat Iverse ad Spread of Gaussia Fiboacci Circulat Matrices I this sectio let A r Circ( G r be a Gaussia Fiboacci circulat matrix First we give the determiat equatio of the matrix A r AfterwardsweprovethatA r is a ivertible matrix for >2adthewefidtheiverse of the matrix A r Obviouslywhe2 r0or A r is also a ivertible matrix Theorem Let A r Circ ( G r be a Gaussia Fiboacci circulat matrix The we have det A r [( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r ] G r ( G r 2 G r is the (r th Gaussia Fiboacci umber Proof I the case >let G r2 Δ 0 0 ( c c c 0 c c 0 c 0 ( ( G r G r 0 0 G r 0 ( G 3 r G r 0 0 Θ G r ( d G 0 r G r 0 0 G r ( (6 (7

3 Abstract ad Applied Aalysis 3 be two matrices; the we have We have ΔA r Θ ( ( f r G r G r3 G r2 0 f r a a 4 a b c 0 0 d b c b f r G rk ( G r G r G r (k (8 det A r [( G r2 G r 2 ( G r 2 (G rk2 G r2 G rk ( G (k r G r ] G r (2 f r ( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r G r a G r G r2 G r (9 Theorem 2 Let A r Circ ( G r be a Gaussia Fiboacci circulat matrix If >2theA r is a ivertible matrix Proof We discuss the sigularity of the matrix A r Whe 3 i Theorem we have det A r ( G r2 G r3 (G 2 r G rg r2 0;heceA r is ivertible I the case >3siceG r (( iβα r (iα β r /(α β αβ αβ letεexp(2πi/; wecagetthatthe eigevalues of A r We obtai a 4 G r4 G r2 G r3 a 3 G r3 G r2 G G r2 r b G r c G r G r f(ε k j G rj (ε k j [( iβ α rj (iα β rj ](ε k j α β j α β [( iβ ( α α r αε k det Δ det A r det Θ f r ( G r 2 [( G r2 G r 2 ( G r 2 (G rk2 G r2 G rk ( G (k r G r ] G r (0 (iα ( β β r βε k ] [αr β r (α r β r i α β ε k ε 2k αr β r (α r β r i ε k ε 2k αr β r (α r β r i ε k ε k ε 2k αr β r (α r β r i ε k ε 2k ε k ] while det Δ( ( ( 2/2 det Θ ( ( ( 2/2 ( G r (G r G r ε k ε k ε 2k (2 (3

4 4 Abstract ad Applied Aalysis Sice G F if letr 0 ε k cos θisi θ θ2kπ/ad 0<θ<2πThe x G r (G r G r ε k [F r F r (F r F r cos θ (F r F r si θ] [F r F r (F r F r si θ (F r F r cos θ] i (4 We assume that Re(x F r F r (F r F r cos θ (F r F r si θ ad Im(x F r F r (F r F r si θ (F r F r cos θ Now we prove that Re(x 0or Im(x 0for ε k ε 2k 0 For the Fiboacci sequece {F }whe> F is a icreasig sequece ad F r F r F r F r F r F r If si θ>0 cos θ>0 Im (x <0; If si θ<0 cos θ<0 Re (x <0; If si θ>0 cos θ<0 Im (x <0; If si θ<0 cos θ>0 Re (x <0 (5 For i jwehave c ij 2 b ik b kj b ii b i j b iib ij (G (G r G r r G r i j ( G r i j (G ( G r r G r i j ( G r i j 0 (9 Hece we verify BB I 2 I 2 is a ( 2 ( 2 idetity matrix Similarly we ca verify B BI 2 Thus the proof is completed Theorem 4 Let A r Circ ( G r ( > 2be a Gaussia Fiboacci circulat matrix The we have A r f r Circ ( 2 (G r2 i hg r i (G r G r i ( G r i h It is verified that whe si θ0or cos θ0 x0whe r0theargumetsfor G r (G r G r ε k 0are similar Hece G r (G r G r ε k 0for ay ε k (k 2 ;thatisf(ε k 0 (k 2 while f( G r (G r G r G r2 G r2 0By Lemma i [9] the proof is completed Lemma 3 Let the matrix B[b ij ] 2 ij be of the form G { r G r i j b ij G { r G r i j { 0 otherwise; the the iverse B [b ij ] 2 ij of the matrix B is equal to b ij { (G r G r i j {( G r i j i j { 0 i < j (6 (7 Proof Let c ij 2 b ikb kj Obviouslyc ij 0for i<ji the case ijweobtai c ii b ii b ii ( G r G r (8 2 (G r i hg r i (G r G r i ( G r i G r3 hg r2 G r (G r3 hg r2 (G r G r ( G r 2 (G r3 hg r2 (G r G r 2 ( G r 3 (G r3 hg r2 (G r G r 3 ( G r 2 f r ( G r2 G r 2 (20 h G r2 (2 (G rk2 G r2 G rk ( G (k r G r G r (22

5 Abstract ad Applied Aalysis 5 Proof Let f r x 3 x 4 x 0 y 3 y 4 y Θ ( (23 d ( x i f r G r3 i (G r2 / G r2 i G r2 i f r (i34 y i G r3 i (G r2 / G r2 i f r (i34 We have f r G rk ( G r G r G r f r ( G r2 G r 2 (k (G rk2 G r2 G rk ( G (k r G r G r (24 ΔA r Θ Θ 2 Λ B (25 Λdiag( f r is a diagoal matrix ad Λ Bis the direct sum of Λ ad B IfwedeoteΘΘ Θ 2 thewe obtai A r Θ(Λ B Δ (26 Sice the last row elemets of the matrix Θ are 0 y 3 y 4 y y ByLemma 3 ifa r Circ(u u 2 u the its last row elemets are give by the followig: u 2 G r2 C ( 2 f r f r u 3 C ( f r u 4 C (2 C ( f r f r u 5 C (3 C (2 C ( f r f r f r Let C (j u C ( 2 C ( 3 C ( 4 f r f r f r u C ( 2 C ( 3 f r f r f r j j (G r3j i (G r2 / G r2j i (G r G r i δ jr ( G r i (27 (μ 2r i (μ r i (j2 2 (28 we have C (2 C ( 2 δ 2r (μ r i (μ 2r i δ r μ 2r G r3 (G r2 / G r2 ( G r 2 (G r G r δ r (μ 2r 2 μ r C ( C (j2 j2 C ( 3 δ 2r (μ r i (μ 2r i 3 δ 3r (μ r i (μ 2r i (G r2 i (G r2 / G r i (μ r i δ r(μ r 3 (μ 2r 2 (μ 2r i (G r2 i (G r2 / G r i (μ r i (μ 2r i C (j C (j δ j2r (μ r i (μ 2r i j δ jr (μ r i (μ 2r i j δ jr (μ r i (μ 2r i (G r4 (G r2 / G r3 (μ r j (μ 2r j (G r3 (G r2 / G r2 (μ r j (μ 2r j2

6 6 Abstract ad Applied Aalysis A r (G r3 (G r2 / G r2 (μ r j (μ 2r j (G r3 (G r2 / G r2 (μ r j We ca get Circ ( C( 2 (μ 2r j2 (j2 4 C ( 3 f r (29 G r3 hg r2 G r (G r3 hg r2 (G r G r ( G r 2 (G r3 hg r2 (G r G r 2 ( G r 3 (G r3 hg r2 (G r G r 3 ( G r 2 (30 C ( 2 (G r2 / f r C( C ( C(2 f r f r h G r2 f r C(3 C (2 C ( f r C( 2 C ( 3 C ( 4 f r f r ( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r G r (3 Circ ( f r Circ ( 2 (G r2 i (G r2 / G r i (μ r i (μ 2r i G r2 2 δ 2r (μ r i (μ 2r i δ r μ 2r δ r (μ 2r 2 μ r δ r (μ 2r 3 (μ r 2 δ r (μ 2r 2 (μ r 3 Lemma 5 (see [23] Let {F } be the Fiboacci sequece we ca have (i j F rj F r2 F r2 (ii j F2 rj F rf r F r F r Lemma 6 Let {G } be the Gaussia Fiboacci sequece we ca have (i j G rj G r2 G r2 (ii j G2 rj G rg r G r Proof By Lemma 5wecaobtai j G rj G r2 G r (F r if r (F r2 if r (F r if r 2 2 (G r2 i hg r i (G r G r i ( G r i h (G r i hg r i (G r G r i ( G r i j F rj i( j F rj F r F r (F r2 F r2 i(f r F r (F r2 if r (F r2 if r G r2 G r2 (32

7 Abstract ad Applied Aalysis 7 Accordig to G G G wehave j G 2 rj G2 r G2 r2 G2 r3 G2 r (G r2 G r G r2 (G r3 G r3 (G r4 G r2 (33 By (6 i [3] we have s (A r (2 [F r (F r F r F r (F r F r2 ] /2 (37 F r is the (r th Fiboacci umber ad α ( 5/2 β ( 5/2 G r (G r G r G r G r G r Theorem 7 Let A r Circ ( G r be a Gaussia Fiboacci circulat matrix; the s(a r (2[F r (F r F r F r (F r F r2 ] /2 (34 F r is the (r th Fiboacci umber ad s(a r is the spread (see [3] of A r Proof From Defiitio 4 i [3]adLemma 5weacquire A r F (( 2 G r2 2 G r 2 /2 ( (α β 2 [(αr β r 2 (α r β r 2 ( (α β 2 [ [ (α r2 β r2 2 (α r β r 2 (α r β r 2 (α r β r 2 ] j 2 (α rj β rj 2 2(α r β r 2 (α r β r 2 (α r β r 2 ] ] (2 (F r F r F r F r (F 2 r F2 r /2 (35 From the elemets i A r wegeta ii sotra r (F r if r ;the 2 A r 2 F 2 tr A r 2 4(F r F r F r F r 2(F 2 r F2 r 2 (F2 r F2 r 4(F r F r F r F r 2(F 2 r F2 r 2[F r (F r F r F r (F r F r2 ] /2 /2 (36 3 Determiat Iverse ad Spread of Gaussia Fiboacci Left Circulat Matrices I this sectio let A r LCirc( G r2 G r be a Gaussia Fiboacci left circulat matrices By usig the obtaied coclusios we give a determiat formula for the matrix A r ad prove that A r is a ivertible matrix for >2for ay positive iteger Theiverseadtheupper boud for spread of the matrix A r arealsopreseted Accordig to Lemma 2 i [9] ad Theorems 2 ad4 we ca obtai the followig theorems Theorem 8 Let A r LCirc (G r be a Gaussia Fiboacci left circulat matrix; the we have det A r ( ( ( 2/2 [( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r ] G r ( G r 2 G r is the (r th Gaussia Fiboacci umber (38 Theorem 9 Let A r LCirc (G r be a Gaussia Fiboacci left circulat matrix; if > 2theA r is a ivertible matrix Theorem 0 Let A r LCirc (G r (>2be a Gaussia Fiboacci left circulat matrix; the we have A r f r LCirc ( 2 (G r2 i hg r i (G r G r i ( G r i (G r3 hg r2 (G r G r 3 ( G r 2 (G r3 hg r2 (G r G r 2 ( G r 3

8 8 Abstract ad Applied Aalysis 2 (G r3 hg r2 (G r G r ( G r 2 G r3 hg r2 G r h (G r i hg r i (G r G r i ( G r i Sice G F if ad A r F (2(F rf r F r F r (F 2 r F2 r /2 ([F r (F r F r F r (F r F r ] /2 (43 by (6 i [3] we ca get the upper bouds for A r as the above easily h G r2 f r ( G r2 G r 2 (G rk2 G r2 G rk (39 (40 4 Determiat Iverse of Gaussia Fiboacci g-circulat Matrices I this sectio let A gr g-circ( G r be a Gaussia Fiboacci g-circulat matrices A determiat formula for the matrix A gr ad the iverse A gr for >2 whe ( g are obtaied as follows From Lemmas 3 ad 4 i [9] ad Theorems 2 ad4 we deduce the followig results Theorem 2 Let A gr g- Circ ( G r be a Gaussia Fiboacci g-circulat matrix; the we have ( G (k r G r G r Theorem Let A r LCirc ( G r ( > 2 be a Gaussia Fiboacci left circulat matrix; the the upper bouds for the spread of A r are s(a r (2[F r (F r F r F r (F r F r ] 2 [(F r2 F r2 2 (F r F r 2 /2 ] ( is odd s(a r (2[F r (F r F r F r (F r F r ] 8 [(F r F r 2 (F r F r 2 /2 ] ( is eve (4 Proof From the elemets i A r ad Lemma 6 if is odd the trace of A r is tr A r j G rj G r2 G r2 ;ifis eve tr A r 2( G r3 G r5 G r 2[ (G rj 3 G rj 2 ] j4 [ ] 2(G r G r (42 det A gr det Q g [( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r ] G r ( G r 2 G r is the (r th Gaussia Fiboacci umber (44 Theorem 3 Let A gr g- Circ ( G r be a Gaussia Fiboacci g-circulat matrix ad (g ;if >2 the A gr is a ivertible matrix Theorem 4 Let A gr g- Circ ( G r (>2be a Gaussia Fiboacci g-circulat matrix ad (g ;the A gr [ f r Circ ( 2 h 2 (G r2 i hg r i (G r G r i ( G r i (G r i hg r i (G r G r i ( G r i

9 Abstract ad Applied Aalysis 9 5 Coclusio G r3 hg r2 G r (G r3 hg r2 (G r G r ( G r 2 (G r3 hg r2 (G r G r 2 ( G r 3 (G r3 hg r2 (G r G r 3 ( G r 2 ]Q T g h G r2 f r ( G r2 G r 2 (G rk2 G r2 G rk ( G (k r G r G r (45 (46 I this paper the explicit determiats ad the iverse matrices of Gaussia Fiboacci circulat type matrices are preseted Furthermore we give the upper bouds for the spread o Gaussia Fiboacci circulat ad left circulat matrices respectively The reaso why we focus our attetios o circulat type matrices is to explore the applicatio of it i the related field O the basis of existig applicatio situatio [ 8] we will develop solvig itegrable system Hamiltoia structure ad itegral equatios Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Ackowledgmets The research is supported by the Developmet Project of Sciece & Techology of Shadog Provice (Grat o 202GGX05 ad the AMEP of Liyi Uiversity Chia Refereces [] J Weiss Factorizatio of the (2 -dimesioal BLP itegrable system by the periodic fixed poits of its Bäcklud trasformatios Physics Letters A vol60o2pp [2] B A Kupershmidt ad G Wilso Modifyig Lax equatios ad the secod Hamiltoia structure Ivetioes Mathematicaevol62o3pp [3]AUOKisiselThe Hamiltoia Structure of Discrete KP Equatios 2000 [4] R H Cha H W Su ad W F Ng Circulat precoditioers for ill-coditioed boudary itegral equatios from potetial equatios Iteratioal Joural for Numerical Methods i Egieerig vol 43 o 8 pp [5] K Malekejad ad M Rabbai Numerical solutio for the Fredholm itegral equatio of the secod kid with Toeplitz kerels by usig precoditioers Applied Mathematics ad Computatiovol80opp [6] I Gohberg M Hake ad I Koltracht Fast precoditioed cojugate gradiet algorithms for Wieer-Hopf itegral equatios SIAM Joural o Numerical Aalysis vol3o2pp [7] H Cai A fast solver for the Hilbert-type sigular itegral equatios based o the direct Fourier spectral method Joural of Computatioal ad Applied Mathematics vol250o4pp [8] M É Abramya Justificatio of the covergece of the method of rectagles for a complete sigular itegral equatio with cotiuous coefficiets o the circle Mathematical Notes vol77o2pp [9] P J Davis Circulat Matrices Joh Wiley & Sos New York NY USA 979 [0] Z L Jiag ad Z X Zhou Circulat Matrices Chegdu Techology Uiversity Publishig Compay Chegdu Chia 999 [] Z Jiag O the miimal polyomials ad the iverses of multilevel scaled factor circulat matrices Abstract ad Applied Aalysisvol204ArticleID526430pages204 [2] Z L Jiag T T Xu ad F L Lu Isomorphic operators ad fuctioal equatios for the skewcirculat algebra Abstract ad Applied Aalysisvol204ArticleID48948pages204 [3] J Li Z Jiag ad F Lu Determiats orms ad the spread of circulat matrices with Triboacci ad geeralized Lucas umbers Abstract ad Applied Aalysis vol 204 ArticleID pages 204 [4] A Bose R S Hazra ad K Saha Spectral orm of circulattype matrices Joural of Theoretical Probability vol 24 o 2 pp [5]CErbasadMMTaik GeeratigsolutiostotheNquees problem usig 2-circulats Mathematics Magazie vol 68 o 5 pp [6] YKWuRZJiaadQLi g-circulat solutios to the (0 matrix equatio A m J Liear Algebra ad Its Applicatios vol 345 o 3 pp [7] E Ngodiep S Serra-Capizzao ad D Sesaa Spectral features ad asymptotic properties for g-circulats ad g- Toeplitz sequeces SIAM Joural o Matrix Aalysis ad Applicatiosvol3o4pp /0 [8] A Cambii A explicit form of the iverse of a particular circulat matrix Discrete Mathematicsvol48o2-3pp [9] Z L Jiag Y P Gog ad Y Gao Ivertibility ad explicit iverses of circulat-type matrices with k-fiboacci ad k- Lucas umbers Abstract ad Applied Aalysisvol204Article ID pages204

10 0 Abstract ad Applied Aalysis [20] X Jiag ad K Hog Exact determiats of some special circulat matrices ivolvig four kids of famous umbers Abstract ad Applied Aalysis vol204articleid pages 204 [2] D Bozkurt ad T Tam Determiats ad iverses of circulat matrices with Jacobsthal ad Jacobsthal-Lucas Numbers Applied Mathematics ad Computatiovol29o2pp [22] Z Jiag J Yao ad F Lu O skew circulat type matrices ivolvig ay cotiuous Fiboacci umbers Abstract ad Applied Aalysisvol204ArticleID483020pages204 [23] SSheJCeadYHao Othedetermiatsadiversesof circulat matrices with Fiboacci ad Lucas umbers Applied Mathematics ad Computatio vol27o23pp [24] Z-L Jiag ad D Li The ivertibility explicit determiats ad i verses of circulat ad left circulat ad g-circulat matrices ivolvig ay cotiuous Fiboacci ad Lucas umbers Abstract ad Applied Aalysis ArtID93454pages 204 [25] A F Horadam Further appearece of the Fiboacci sequece The Fiboacci Quarterlyvolo4pp [26] A İpek ad K Arı O Hesseberg ad petadiagoal determiats related with Fiboacci ad Fiboacci-like umbers Applied Mathematics ad Computatio vol229pp

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