Annales Mathematicae et Informaticae 49 08 pp 75 84 doi: 009/ami080900 http://amiuni-eszterhazyhu On identities with multinomial coefficients for Fibonacci-Narayana sequence Taras Goy Vasyl Stefany Precarpathian National University, 7608 Ivano-Franivs, Uraine tarasgoy@yahoocom Submitted August 4, 07 Accepted September 5, 08 Abstract In this paper we study some families of Toeplitz-Hessenberg determinants the entries of which are Fibonacci-Narayana or Narayana s cows numbers This leads to discover some identities for these numbers In particular, we establish connection between Fibonacci-Narayana numbers with Fibonacci and tribonacci numbers We also present new formulas for Fibonacci-Narayana numbers via recurrent determinants of four-diagonal matrix Keywords: Fibonacci-Narayana sequence, Narayana s cow sequence, Toeplitz- Hessenberg matrix, Fibonacci sequence, tribonacci sequence MSC: AMS classification B9, 5B05 Introduction The Fibonacci sequence {F n } n 0 is defined by the initial values F 0 = 0, F = and the recurrence relation F n = F + F n, where n Among the several generalizations of Fibonacci numbers, some of the best nown are the tribonacci sequence {t n } n 0 and the Fibonacci-Narayana sequence or Narayana s cows sequence {b n } n 0, which are defined by the following thirdorder recurrence relations: for n t n = t + t n + t n, t 0 = t = 0, t =, b n = b + b n, b 0 = 0, b = b =, 75
76 T Goy There are large number of sequences indexed in OEIS [0], being in this case {F n } n 0 = {0,,,,, 5, 8,,, 4, 55, 89, 44,, 77, } : A000045 {t n } n 0 = {0, 0,,,, 4, 7,, 4, 44, 8, 49, 74, 504, 97, } : A00007 {b n } n 0 = {0,,,,,, 4, 6, 9,, 9, 8, 4, 60, 88, 9, } : A00090 The Fibonacci-Narayana sequence was introduced by the Indian mathematician Narayana in the 4th century, while studying the following problem: A cow produces one calf every year Beginning in its fourth year, each calf produces one calf at the beginning of each year How many cows are there altogether after, for example, 0 years? This problem can be solved in the same way that Fibonacci solved its problem about rabbits [6] There has been considerable recent interest in the Fibonacci-Narayana sequence and its generalizations see [,,, 4, 5, 9, ] for more details For instance, Didivsa and St opochina [4] proved some basic properties of Fibonacci-Narayana numbers Biglici [] defined a generalized order- Fibonacci-Narayana sequence and by using this generalization and some matrix properties, established some identities related to Fibonacci-Narayana numbers Flaut and Shpaivsyi [5] studied some properties of generalized and Fibonacci quaternions and Fibonacci-Narayana quaternions Ramírez and Sirvent [9] defined the -Narayana sequence of integer numbers and studied recurrence relations and some combinatorial properties of these numbers, and of the sum of their first n terms These authors also established some relations between the -Narayana sequence and determinants of one type of Hessenberg matrix The purpose of this paper is to study Fibonacci-Narayana numbers We investigate some families of Toeplitz-Hessenberg determinants the entries of which are Fibonacci-Narayana numbers This leads to discover some identities for these numbers In particular, we establish connection between Fibonacci-Narayana numbers with Fibonacci and tribonacci numbers We also present new formulas for Fibonacci-Narayana numbers via recurrent determinants of four-diagonal matrix Toeplitz-Hessenberg matrices and determinants A Toeplitz-Hessenberg matrix is an n n matrix of the form a a 0 0 0 0 a a a 0 0 0 a a a 0 0 M n a 0 ; a, a,, a n =, a a n a n a a 0 a n a a n a a where a 0 0 and a 0 for at least one > 0 So a ij = 0 for j > i + Several mathematical objects may be represented as determinant of such matrices see, for example, [7] and the references given there
On identities with multinomial coefficients for Fibonacci-Narayana sequence 77 Expanding the determinant detm n according to the first row repeatedly, we obtain the recurrence detm n = n a 0 a detm n, = where, by definition, detm 0 The following result is nown as Trudi s formula [8] This gives the multinomial extension for detm n Lemma Let n be a positive integer Then detm n = a 0 n s+ +sn s + + s n a s s,, s as asn n, n s,,s n where the summation is over integers s i 0 satisfying s + s + + ns n = n, and s + + s n = s + + s n! s,, s n s! s n! is the multinomial coefficient Example It follows from that detm = a 0 0 a + a 0 a a + a 0 a, 0, 0,, 0 0, 0, = a a 0 a a + a 0a ; 4 detm 4 = a 0 0 a 4 + a 0 a 4, 0, 0, 0,, 0, 0 a + a 0 a a, 0,, 0 + a 0 a + a 0 a 4 0,, 0, 0 0, 0, 0, = a 4 a 0 a a + a 0a a + a 0a a 0a 4 Throughout this paper, we denote det±; a, a,, a n = det M n ±; a, a,, a n Connection formulas between the Fibonacci-Narayana numbers and Fibonacci numbers The next theorem gives relationship between Fibonacci-Narayana numbers and Fibonacci numbers via the Toeplitz-Hessenberg determinants
78 T Goy Theorem For all n, det; b, b,, b = n F, det; b 0, b,, b n = F n Proof We will prove formula using induction on n The other proof follow similarly, so we omit it for interest of brevity For simplicity of notation, we write D n instead of det; b, b,, b Clearly, formula wors, when n = and n = Suppose it is true for all positive integers n, where n Using recurrence, we have D n = n i b i D n i = b D b D n + b 5 D n + = D D n + D n + n i b i + b i 4 D n i i=4 n i b i + b i 5 + b i 7 D n i i=4 = D D n + D n + i b i D n i i= n + i+ b i D n i + i= = D D n + D n + n i+ b i D n i i b i D n i + b D n b D n n + i b i D n i b D n D n = D D n + D n D + D n D n + D n D n D n = D n D n Using the induction hypothesis and the definition of the Fibonacci sequence, we obtain D n = n F n n F n 4 = n F n + F n 4 = n F Consequently, the formula is true for n formula wors for all positive integers n Therefore, by induction, the
On identities with multinomial coefficients for Fibonacci-Narayana sequence 79 4 Connection formula between the Fibonacci-Narayana numbers and tribonacci numbers The next theorem gives relationship between Fibonacci-Narayana numbers and tribonacci numbers via the Toeplitz-Hessenberg determinants Theorem 4 For all n, det ; b 0, b,, b = t n 4 Proof We will prove formula 4 using induction on n det ; b 0, b,, b = n D n, where D n = det; b 0, b,, b It is easily seen that Clearly, formula 4 wors, when n = and n = Suppose it is true for all n, where n Using recurrence, we have n D n = i b i D n i = b 0 D + b D n b D n + n i b i + b i 4 D n i i=4 = D n D n + i+ b i D n i + = D n D n + i= n i+ b i D n i i+ b i D n i b 0 D n + b D n D n = D n D n D + D n D n = D + D n D n Thus, det; b 0, b,, b = n t + = t + t n + t n = t n Consequently, the formula 4 is true for n formula wors for all positive integers n t n n t n n Therefore, by induction, the 5 Some Toeplitz-Hessenberg determinants with Fibonacci-Narayana entries In this section, we evaluate det±; a, a,, a n with special Fibonacci-Narayana entries a i
80 T Goy Theorem 5 Let n, except when noted otherwise Then det; b 0, b,, b = + n+, det; b, b,, b n = + n/, det ; b, b,, b n = n i 8 i, 5 i i=0 det; b, b,, b n+ = 0, n 4, n i + det; b, b 4,, b n =, n i i=0 i det ; b, b 4,, b n =, 4n i i=0 det; b, b 4,, b n+ = n + n /, n, det; b, b 5,, b n+ = 0, n 4, + n det; b 4, b 5,, b n+ =, det; b 4, b 6,, b n+ =, n, n+ n + i det; b 5, b 6,, b n+4 =, + i i=0 det; b 5, b 7,, b n+ = n +, n, det; b 6, b 8,, b n+4 = n + n + 4/, where m = m!!m! is the binomial coefficient and is the floor function Proof We will prove only 5, the other ones can be proved in the same way Obviously, det ; b, b,, b n = n D n, where D n = det; b, b,, b n, When n = and n =, the formula holds Assuming 5 to hold for all, we prove it for n Using, we have n D n = i b i D n i = b D + b D n b D n + n i b i + b i 4 D n i i=4 = D + D n D n + i+ b i D i + i= n i+ b i D n i
On identities with multinomial coefficients for Fibonacci-Narayana sequence 8 Thus, = D + D n D n + i+ b i D i D n + D n D n = D + D n D n D D n + D n D n = D D n n = n n 4 n 4 n D n = n n 5 = Let n m Then n = From 5, using well-nown formula n n n + =, 5 we have D n = n = Let n = m Then Therefore, n = n n = Now, using 5, we have D n = n = n n det ; b, b,, b n = n+ n = n Since the formula 5 holds for n, it follows by induction that it is true for all positive integers n
8 T Goy The Trudi formula, taen together with Theorems and 4 yields the following corollary Similarly, one can obtain the multinomial extensions for formulas from Theorem 5 Corollary 5 The following formulas hold: σ p n sb s 4 bs n = F n, n, 54 s +s + +ns =n s +s + +ns n=n s +s + +ns =n σn p n sb s bs bs bsn = n F, n, 55 p n sb s bs bs = t n, n, 56 where p n s = s + +s n s,,s n is the multinomial coefficient, σn = s + + s n, s i 0, F n and t n are the n-th Fibonacci and tribonacci numbers, respectively Example 5 It follows from 54, 55, and 56, respectively, that b b b 6 b 4 + b 0 = F 6, b 5 4b b + b b 5 + b b b b 7 b b 5 + b 9 = F 4, b + b b 4 + b b + b 6 = t 7 6 Fibonacci-Narayana determinants In this section, we prove two formulas expressing Fibonacci-Narayana numbers b i with even odd subscripts via recurrent determinants of four-diagonal matrix of order n Let P n and Q n denote the n n four-diagonal matrices and P n = Q n = b 0 b 4 0 b 5 b 0 b 6 b 7 b 5 0 b 8 0 b n b n 5 0 b n b b n 0 b 0 b 0 b 4 b 0 b 5 b 6 b 4 0 b 7 0 b n 4 b n 6 0 b n b n b n 4 0
On identities with multinomial coefficients for Fibonacci-Narayana sequence 8 Theorem 6 For all n, b n = b = detp n, 6 b i b i detq n 6 Proof We prove only formula 6, the formula 6 one can be proved similarly We use induction on n Since detp = = b and detp = = b 4, the result is true when n = and n = Assume it true for every positive integer < n Expanding detp n by the last row, we have b n = = b i b i = b n + b bn b n detp n + b b n b n 4 detp n n b n b n Therefore, the result is true for every n n b i + b b n b n 4 b i Acnowledgements I would lie to than the referee for a very careful reading of the manuscript and several useful suggestions References [] Allouche, J-P, Johnson, T, Narayana s cows and delayed morphisms, In: Articles of rd Computer Music Conference JIM 96, France, Vol 4 996, 7 [] Ballot, C, On a family of recurrences that includes the Fibonacci and the Narayana recurrence, https://arxivorg/abs/70404476v [] Biglici, G, The generalized order- Narayana cow s numbers, Math Slovaca, Vol 664 06, 795 80 https://doiorg/055/ms-05-08 [4] Didivsa, TV, St opochina MV, Properties of Fibonacci-Narayana numbers, In the World of Mathematics, Vol 9 00, 9 6 in Urainian [5] Flaut, C, Shpaivsyi, V, On generalized Fibonacci quaternions and Fibonacci- Narayana quaternions, Adv Appl Clifford Algebras, Vol 0, 67 688 https://doiorg/0007/s00006-0-088- [6] Koshy, T, Fibonacci and Lucas Numbers with Applications, John Wiley, New Yor, 00 https://doiorg/000/97880067
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