Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp
Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No., pp. 5 TRIBONAI NUMBERS WITH INDIES IN ARITHMETI PROGRESSION AND THEIR SUMS NURETTIN IRMAK AND MURAT ALP Received May, 0 Abstact. In this pape, we give a ecuence elation fo the Tibonacci numbes with indices in aitmetics pogession, ft ns g fo 0 s < n. We find sums of ft n g fo abitay intege via matix methods. 000 Mathematics Subject lassification B9; 0 Keywods Tibonacci numbes, sums, matix methods. INTRODUTION Fo n ; the Tibonacci numbes ae defined by T n D T n T n T n (.) with initial conditions T 0 D 0;T D T D The Binet fomula of the Tibonacci numbes is given by, T n D c n c ˇn c n D c n c 5ˇn c 6 n whee c D. ˇ/. / ;c D.ˇ /.ˇ / ;c D. /. ˇ/ ;c D ;c 5 D ˇ ˇ ;c 6 D and ;ˇ; ae the oots of the equation x x x D 0 The Tibonacci sequence is a special genealization of Fibonacci sequence. Many authos studied the Tibonacci sequence and its vaious popeties. Fo example, in [], the autho found a fomula fo the Tibonacci numbes by using analytic methods. In [], the autho studied elationships between thid ode sequences with diffeent initial conditions. And in [0], the autho investigated the Tibonacci numbes De moive-type identities. In [5], the autho found some fomulas and identities and geneating matices fo the sequences ft n g;ft n g and thei sums. In [9], one can find the well-known identities fo Tibonacci sequences. Fo example, c 0 Miskolc Univesity Pess
6 NURETTIN IRMAK AND MURAT ALP the geneating matix of Tibonacci sequence is given by, Q n D n 0 0 5 D T n T n T n T n T n T n T n T n 0 0 T n T n T n T n Fo moe details, we efe to [,,, ]. The matix method is vey useful method in ode to obtain some identities fo special sequences. Fo example, using matix methods, the authos obtained some identities fo vaious special sequences (see [,, 6 8]). In this study, we will conside cetain subsequences of the Tibonacci sequence of fom ft ns g fo abitay integes ;s with 0 s <. Afte, we will find some fomulas fo sums of the sequence ft n g by using matix methods. Now we ae giving a lemma and a theoem fo the subsequence of the Tibonacci sequences to detemine the ecuence elation. Lemma. Let ; ˇ and ae oots of x x x D 0 Then, fo any positive intege ;. ˇ /;. ˇ/. /.ˇ/ and. ˇ/ ae always integes. Poof. We will use the induction method. Obviously, ˇ D ; ˇ ˇ D and ˇ D Now, suppose that. n ˇn n /;. ˇ/ n. / n.ˇ/ n and. ˇ/ n ae the integes fo 0 < n. Then, ˇ. ˇ / D ˇ and since ˇ ˇ D, 5. ˇ / ˇ. ˇ ˇ/. ˇ/ ˇ. ˇ / D ˇ. ˇ/ Finally, we obtain ˇ. /. ˇ/ D ˇ ˇ. ˇ/ ˇ ˇ D ˇ ˇ ˇ Since. ˇ /; ˇ and ˇ ae the integes, ˇ is the intege. Similaly, it can be shown that the tems of. ˇ/. /.ˇ/ and. ˇ/ ae the integes. Theoem. Fo n > and the integes ; s such that 0 s <, T ns D ˇ T.n /s. ˇ/. /.ˇ/ T.n /s. ˇ/ T.n /s ; (.) whee ; ˇ and ae the oots of chaacteistic polynomial of the Tibonacci sequence x x x
TRIBONAI NUMBERS WITH INDIES IN ARITHMETI PROGRESSION Poof. In ode to pove the claim, we will use the Binet fomula of the Tibonacci sequence. If we evaluate the ight hand side of (.), then ˇ T.n /s. ˇ/. /.ˇ/ T.n /s. ˇ/ T.n /s D ˇ c ns c ˇns c ns. ˇ/. /.ˇ/ c.n /s c ˇ.n /s.n c /s. ˇ/ c.n /s c ˇ.n /s c.n /s ; whee c D. ˇ/. / ;c D.ˇ /.ˇ / and c D If we simplify the. /. ˇ/ above equation, then we obtain ˇ T.n /s. ˇ/. /.ˇ/ T.n /s. ˇ/ T.n /s D c.n/ c ˇ.n/ c.n/ D T ns Thus the poof is complete. When D in the above theoem, we obtain the well-known Tibonacci ecuence.. SUMS OF TERMS OF THE SUBSEQUENE ft n g In this section, we will compute sums of the Tibonacci numbes ft n g and thei sums by matix methods. nx S n; D T k (.) kd0 whee is an intege, and define matices F and G n; as shown, 0 0 0 F D 6./././ 0 0 0 5 0 0 0 and T T 0 0 0 S n; T.n/./ T n./ T.n /./ T n G n; D 6 S n ; T n./ T.n /./ T.n /./ T.n / 5 ; S n ; T.n /./ T.n /./ T.n /./ T.n /
8 NURETTIN IRMAK AND MURAT ALP whee./ D ˇ ;./ D. ˇ/.ˇ/. / and./ D. ˇ/ Fo n ; the sequence ft g is defined as follows whee initial conditions T 0 D T D 0;T D Theoem. Fo n ; T n D T n T n T n ; (.) T F n T F n D G n; Poof. The poof follows fom the induction method. Afte some computations, the eigenvalues of matix F ae ;ˇ; and. Define two matices L and W as follows 0 0 0 0 0 0 L k D 6 0 k 0 0 0 0 ˇk 0 5 and W k ˇk k D././ 6 k ˇk k 0 0 0 k././ 5 and Theoem. If n > 0; then S n; D././ nt.n/./././ T n./ T.n / T T o Poof. Since ;ˇ ae diffeent zeo, then detl k 0 One can check that T F n W D T W L n T F n W D T W L n If we sum both equations side by side, we obtain that T F n T W D W T L n k T By Theoem, we deduce Equating the.; / poof. F n G n; W D W T L n k T Ln k Ln k elements of each sides of the above equation completes the In the above theoem, we give a fomula fo sum of the tems of the sequence ft n g fo abitay and fo the geneating matix of the sums. Now we define two
TRIBONAI NUMBERS WITH INDIES IN ARITHMETI PROGRESSION 9 new matices K and R n; in ode to give a Binet like-fomula fo the sum of the tems of the sequence ft n g,././././ K D 6 0 0 0 0 0 0 5 0 0 0 and whee X n; D R n; D 6 S n; X n; Y n; S n; S n; X n ; Y n ; S n ; S n ; X n ; Y n ; S n ; S n ; X n ; Y n ; S n ;././ Y n D and S n; was defined by (.) Theoem. Fo n ; S n;././ 5 S n ; S n ; ; S n; S n ; ; T K n T Kn D R n; (.) whee ft g is the Tibonacci sequence and the sequence ft g is defined as in (.). Poof. The poof follows fom the induction method. Now we define two matices and D as follow ˇ D 6 ˇ ˇ 5 and D D 6 Theoem 5. Fo n > ; S n; D T 0 0 0 0 ˇ 0 0 0 0 0 0 0 0 5.n/ ˇ.n/. /. ˇ/. /.ˇ /.ˇ /.ˇ /.n/. /. /. ˇ/ T.n/ ˇ.n/. /. ˇ/. /.ˇ /.ˇ /.ˇ /.n/. /. /. ˇ/
0 NURETTIN IRMAK AND MURAT ALP Poof. The chaacteistic equation of the matix K is x./ x./ x././ x D 0 omputing oots of the equation, we obtain ; ˇ; and So the matix K is diagonalizable. It can be shown that and T K n D T D n T Kn Kn D T D n If we sum the both equalities, we obtain T K n T D T D n T Using (.), we deduce that R n; D So, we have the following equation system, T D n T Dn x./ i; x./ i; x./ i; x./ i; D T.n ˇ x./ i; ˇ x./ i; ˇx./ i; x./ i; D T ˇ.n Dn i/ T i /.n i/ T i / ˇ.n x./ i; x./ i; x./ i; x./ i; D T.n i/ T.n i / x./ i; x./ i; x./ i; x./ i; D T T whee R n; D x./ i;j In ode to obtain x./; ; we use ame s method T.n/ T.n/ det6 T ˇ.n/ T ˇ.n/ ˇ ˇ T.n/ T.n/ 5 T T x./ ; D ˇ det6 ˇ ˇ 5 If we expand the fist column of the matix, T.n/ T.n/ det6 T ˇ.n/ T ˇ.n/ ˇ ˇ T.n/ T.n/ 5 T T
D det6 TRIBONAI NUMBERS WITH INDIES IN ARITHMETI PROGRESSION T.n/ T ˇ.n/ ˇ ˇ T.n/ T 5 det and afte some simplifications, we obtain S n; as 6 T.n/ T ˇ.n/ ˇ ˇ T T 5 ; D T x./ ; D S n;.n/ ˇ.n/. /. ˇ/. /.ˇ /.ˇ /.ˇ /.n/. /. /. ˇ/ T.n/ ˇ.n/. /. ˇ/. /.ˇ /.ˇ /.ˇ /.. Deteminantal epesentations.n/. /. /. ˇ/ In this section, we give some elationships between the sequence ft n g, the sum of tems of the sequence and the pemanents of cetain matices. We define n n matices H n; ; Z n; as follows 8 H n; D ˆ< u./ i;j D ˆ./ if i D j;./ if j D i ;./ if j D i ; if i D j ; 0 othewise and Z n; D 0 H n 6 5 0 We pesent some elationships with the following theoem. Theoem 6. Fo n > (i) T peh n; T peh n (ii) T pez n; T pez n ; D T.n/ ; D P n j D T j
NURETTIN IRMAK AND MURAT ALP Poof. (i) If we expand the pemanent of matix H n; accoding to the fist column, then we obtain peh n; D./ peh n;./ peh n ;./ peh n ; Since peh n; and ft n g have the same ecuence elation and initial conditions, the poof is complete. (ii) It can be poven similaly to the fist identity.. OMBINATORIAL REPRESENTATIONS In this pat, we give some combinatoial epesentations of the tems of the sum of subsequence of the Tibonacci sequence and the enties of the n th powe of the companion matix k k k k 0 0 A k.k ;k ;;k k / D 6 5 0 0 0 In [], we can see the following esults; Theoem. The.i;j / enty a.n/ i;j.k ;k ;;k k / in matix A.n/ k.k ;k ;;k k / is given by following fomula D X.t ;t ;;t k / a.n/ i;j.k ;k ;;k k / t j t j t k t t t k t t t k t ;t ;;t k k t k t k k (.) whee the summation is ove nonnegative integes satisfying t t kt k D n i j; and the coefficient in (.) is defined to be if n D i j oollay. Let S n; be the sum of Tibonacci numbes whee the subscipts fom an aithmetic sequence. Then X S n; D./././ ; ; ;. ; ; ; /./. / whee the summation is ove nonnegative integes satisfying D n oollay. Let T n be the n th Tibonacci numbe. Then T n D X t t t./ t./ t t ;t ;t.t ;t ;t /./ t
TRIBONAI NUMBERS WITH INDIES IN ARITHMETI PROGRESSION whee the summation is ove nonnegative integes satisfying t t t D n AKNOWLEDGEMENT The authos would like to thank the anonymous efeee fo a numbe of helpful suggestions. REFERENES [] K. Alladi and V. E. j. Hoggatt, On Tibonacci numbes and elated functions, Fibonacci Q., vol. 5, pp. 5, 9. [] W. Y.. hen and J. D. Louck, The combinatoial powe of the companion matix, Linea Algeba Appl., vol., pp. 6 8, 996. [] E. Kiliç, The genealized ode-k Fibonacci-Pell sequence by matix methods, J. omput. Appl. Math., vol. 09, no., pp. 5, 00. [] E. Kiliç, Sums of genealized Fibonacci numbes by matix methods, As omb., vol. 8, pp., 00. [5] E. Kiliç, Tibonacci sequences with cetain indices and thei sums, As omb., vol. 86, pp., 008. [6] E. Kiliç, The genealized Fibonomial matix, Eu. J. omb., vol., no., pp. 9 09, 00. [] E. Kiliç, A matix appoach fo genealizing two cuious divisibility popeties, Math. Notes, Miskolc, vol., no., pp. 89 96, 0. [8] E. Kiliç and P. Stănică, A matix appoach fo geneal highe ode linea ecuences, Bull. Malays. Math. Sci. Soc. (), vol., no., pp. 5 6, 0. [9] T. Koshy, Fibonacci and Lucas numbes with applications, se. Pue and Applied Mathematics. A Wiley-Intescience Seies of Texts, Monogaphs, and Tacts. New Yok Wiley, 00. [0] P.-Y. Lin, De Moive-type identities fo the Tibonacci numbes, Fibonacci Q., vol. 6, no., pp., 988. []. P. Mcaty, A fomula fo Tibonacci numbes, Fibonacci Q., vol. 9, pp. 9 9, 98. [] H. Minc, Pemanent of.0;/-ciculants, an. Math. Bull., vol., pp. 5 6, 96. [] S. Pethe, Some identities fo Tibonacci sequences, Fibonacci Q., vol. 6, no., pp. 5, 988. Authos addesses Nuettin Imak Niğde Univesity, Mathematics Depatment, 50 Niğde, TURKEY E-mail addess nimak@nigde.edu.t Muat Alp Niğde Univesity, Mathematics Depatment, 50 Niğde, TURKEY E-mail addess muatalp@nigde.edu.t