Some new properties of Fibonacci n-step

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1 South Asian Journal of Mathematics, Vol. 4 ( 3): ISSN - RESEARCH ARTICLE Some new properties of ibonacci n-step Manjusri Basu 1, Monojit Das 1 1 Department of Mathematics, University of Kalyani, Kalyani, W.B., India manjusri basu@yahoo.com Received: Dec--; Accepted: Jane-- *Corresponding author Abstract We search the ibonacci n-step numbers and tabulate these numbers for = 0, ±1, ±2,...,± and n = 1, 2, 3,...,. Then we show some properties for generalizing these numbers for any integral values of and for any n 1. Key Words MSC ibonacci numbers, ibonacci n-step numbers A, B3 1 Introduction The ibonacci numbers are defined by the second-order linear recurrence relation: +1 = + 1 (1.1) with the initial terms 0 = 0, 1 = 1. This identity is called Cassini formula in honor of the well-nown th century astronomer Giovanni Cassini (-) who derived this formula. ibonacci n-step numbers are defined by linear recurrence relation of order n > 1: with n initial terms = n (1.2) 0 = 1 = = (n) n 2 = 0, n 1 = 1. The numbers generated by the equation (1.2) are also nown as the -generalized ibonacci numbers, which are discussed by lores [4] in 7. Equation (1.2) is equivalent to the recurrence relation of three terms = 2 1 n 1. In, L. E. Dicson [3] discussed a long history of generalizations of the ibonacci numbers. In 0, E. P. Miles [6] used equation (1.2) and in 03 Bengamin et. al. [1] briefly discuss a combinatorial interpretation of these n-step numbers. When n = 1 the recurrence relation (1.1) gives the degenerate series 1, 1, 1, 1, 1,.... Citation: Manjusri Basu, Monojit Das, Some new properties of ibonacci n-step, South Asian J Math,, 4(4), 0-0.

2 South Asian J. Math. Vol. 4 No. 3 or n = 2, the recurrence relation (1.1) generates the usual ibonacci numbers. Similarly for small values of n, e.g. n = 3, 4, 5, 6, 7, 8 etc. we obtain Tribonacci, Tetranacci (Quadranacci), Pentanacci (Pentacci), Hexanacci (Esanacci), Heptanacci, Octanacci etc. respectively. or 1, r n = lim (n) exists, called n-anacci constant and is the real root 1 of the equation 1 x n x n 1 x n 2 x 1 = 0, or equivalently x n (2 x) = 1. or even n, there are exactly two real roots, one greater than 1 and one less than 1, and for odd n, there is exactly one real root, which is always 1. Again I. lores [4] proves that lim n r n = 2. In 05, Tony D. Noe et.al. [8] tabulate the values of that yield the prime terms of the ibonacci n-step sequences. Definition 1.1. A prime in the form 2 n 1 is called Mersenne prime. It is well nown that every Mersenne prime appears in ibonacci n-step sequences as 2n. There are only nown Mersenne prime which are listed in Table 1 along with ibonacci n-step number. (4) 86 is the largest nown prime as well as Mersenne prime. Index Mersenne prime As ibonacci n step number (2) (3) (5) (7) () () (1) () (61) (8) (7) (7) (5) (7) (7) 2 1 () (81) () () () (68) () (2) (7) 3874 Table 1 Index Mersenne prime As ibonacci n step number (1) () () (863) (03) (4) (1) (783) (8) (787) (8) (62) (7) (673) (6) () (03) (61) () (38) (6667) () (4) 86 The purpose of this paper is to tabulate the ibonacci n-step numbers for = 0, ±1, ±2, ±3,..., ± and n = 1, 2, 3,...,. Then shows some properties for generalizing these numbers for positive and negetive values of and for n 1. The following is Table 2-: 1

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11 M. Basu, et al : Some new properties of ibonacci n-step 2 Some properties of (1) or n 2, 0 = 1 = = (n) n 1 = 0, n 1 = 1 (initial condition), n = 2 0, n+1 =, n+2 =, n+3 =,... 2n 1 = 2n 1, 2n = 2n 1. (2) or n m( 2), 2n+m = 2n+m a m, where a m satisfy the recurrence relation a m = 2a m m 1, a 0 = 1. (3) or n 1, 1 = 1 and for n 2, (n) 2 = 1. (4) or n 2, (n) (n) n 1 = 2, n 2 = 3 and 2n 2 = 8. (5) or n 3, (n) (n) = 0 for = 3, 4, 5,...; 2n 1 = 4 and 2n 3 = 5. (6) or n m( 4), ( (m 3)n m) = 0, ( (m 2)n) = 0. ( (m 3)n m 1) = 0, ( (m 3)n m 2) = 0,... All these poperties can easily proved by induction using the equation (1.2) and Table 2 - Table. References 1 A. T. Bengamin, J. J. Quinn, The ibonacci numbers-exposed more discretely, Math. Mag. 76 (03), M. Catalani, Polymatrix and generalized polynacci numbers, published electronically at 3 L. E. Dicson, History of the theory of numbers, AMS Chelsea vol. 1,. 4 I. lores, Direct calulation of K-generalized ibonacci numbers, ibonacci Quart., 5(7),-6. 5 T. Koshy, ibonacci and Lucas numbers with applications, John Wiley and Sons, NY, E, p. Miles, Jr., Generalized ibonacci numbers and associated matrices, Amer, Math. Monthly 67 (0), E. W. Weisstein, ibonacci n-step number, published electronically at mathworld.wolfram.com/ibonacci n-step Number.html. 8 Tony D. Noe, Jonathan Vost Post, Primes in ibonacci n-step and Lucas n-step Sequences, Journal of Integer Sequences, vol. 8(05), artcle