A note on closed-form representation of Fibonacci numbers using Fibonacci trees
aa r X i v : . [ m a t h . C O ] F e b A NOTE ON CLOSED-FORM REPRESENTATION OF FIBONACCINUMBERS USING FIBONACCI TREES
INDHUMATHI RAMAN
Abstract.
In this paper, we give a new representation of the Fibonacci numbers. This isachieved using Fibonacci trees. With the help of this representation, the n th Fibonacci numbercan be calculated without having any knowledge about the previous Fibonacci numbers. Introduction
A Fibonacci tree is a rooted binary tree in which for every non-leaf vertex v , the heights ofthe subtrees, rooted at the left and right child of v , differ by exactly one. A formal recursivedefinition of the Fibonacci tree (denoted by F h if its height is h ) is given below. Definition 1.1. F := K , F := K . For h ≥ , F h is obtained by taking a copy of F h − , acopy of F h − , a new vertex R and joining R to the roots of F h − and F h − . Figure 1 shows this construction and a few small Fibonacci trees. F h − F h − F h R R R F F F F Figure 1.
Recursive construction and examples of Fibonacci TreesThe above recursive definition implies that the number of vertices in F h is | V ( F h ) | = | V ( F h − ) | + | V ( F h − ) | + 1. On solving this recurrence relation, we get | V ( F h ) | = f ( h + 2) − f ( i ) is the i th number in the Fibonacci sequence, f (0) = 1 , f (1) = 1 , f ( n ) = f ( n −
1) + f ( n − F h in closed-form by observing the number of vertices at each level of F h . Such a calculation helps us to give a closed-form representation of n th Fibonacci numberfor every n ≥ A closed-form is one which gives the value of a sequence at index n in terms of only one parameter, n itself. . Closed-form representation of Fibonacci numbers
There are several closed-form representations of the Fibonacci numbers. We state a fewbelow. • f ( n ) = (1+ √ n − (1 −√ n n √ It was derived by Binet [3] in 1843, although the result was known to Euler, DanielBernoulli, and de Moivre more than a century earlier. • B ( x ) = ∞ X k =0 b k x k In the above generating function for the Fibonacci numbers’ the value of b k givesthe k th Fibonacci number. However expanding the generating function involves tediouscalculations. • f n = round (cid:16) √ (cid:16) √ (cid:17) n (cid:17) It was also derived by Binet [3] where the function round () rounds the simplifiedexpression up or down to an integer.In this section, we give a simpler closed-form combinatorial representation for Fibonaccinumbers. To do so, we first give a closed-form representation for the number of vertices | V ( F h ) | of F h (the Fibonacci tree of height h ). One of the most powerful methods of solvingequations is called Guess-and-Check - We can use any method to figure out a solution to anequation and then determine whether it is right by substituting it in the equation and checkingthe equation is solved by the guessed solution. This method is particularly powerful when usedwith differential equations, recurrence equations and simultaneous equations - Such a methodis employed in the following lemma to calculate the number of vertices in a particular level of F h and thereafter we sum the number of vertices over the levels to get | V ( F h ) | . Lemma 2.1.
Let F h be a Fibonacci tree of height h and let k be an integer such that ≤ k ≤ h .The number of vertices N ( h, k ) at level k of F h is given by N ( h, k ) = h − k X i =0 (cid:18) kh − k − i (cid:19) Proof : We prove the lemma by induction on k . For k = 0 we have N ( h,
0) = h X i =0 (cid:18) h − i (cid:19) .Using the convention (cid:0) nr (cid:1) = 0 if n < r , we have N ( h,
0) = (cid:0) (cid:1) = 1. This is true since the rootof F h is the only vertex at level 0. Further proceeding, from the recursive definition of F h , we2 VOLUME , NUMBERave N ( h, k ) = N ( h − , k −
1) + N ( h − , k − h − k X i =0 (cid:18) k − h − k − i (cid:19) + h − k − X j =0 (cid:18) k − h − k − j − (cid:19) = h − k X i =0 (cid:18) k − h − k − i (cid:19) + h − k X j =0 (cid:18) k − h − k − j − (cid:19) − (cid:18) k − − (cid:19) = h − k X i =0 (cid:18)(cid:18) k − h − k − i (cid:19) + (cid:18) k − h − k − i − (cid:19)(cid:19) since (cid:18) nr (cid:19) = 0 if r < h − k X i =0 (cid:18) kh − k − i (cid:19) In Step 3 of the above equation, we add and subtract (cid:0) k − h − k − j − (cid:1) for j = h − k . This provesthe lemma. ✷ The number of vertices in any tree is the sum of the vertices at its levels. In particular, | V ( F h ) | = h X k =0 N ( h, k ). Hence we have the following lemma. Lemma 2.2.
Let F h be the Fibonacci tree of height h , then the number of vertices | V ( F h ) | of F h is h X k =0 h − k X i =0 (cid:18) kh − k − i (cid:19) . ✷ The above theorem helps us to derive a closed-form representation for the Fibonacci num-bers. This representation is in contrast to the recurrence relation form, which has certainprevious values of the sequence as parameters. We know that | V ( F h ) | = f ( h + 2) −
1. Equiv-alently f ( n ) = 1 + | V ( F n − ) | . Theorem 2.3.
Let f ( n ) be the n th number in the Fibonacci sequence starting with f (0) = 1 and f (1) = 1 . Then for n ≥ , f ( n ) = 1 + n − X k =0 n − k − X i =0 (cid:18) kn − k − i − (cid:19) Proof : An immediate consequence of Lemma 2.2. ✷ MONTH YEAR 3s an example for Theorem 2.3, we calculate f (4) and f (5). f (4) = 1 + X k =0 2 − k X i =0 (cid:18) k − k − i (cid:19) = 1 + X i =0 (cid:18) − i (cid:19) + X i =0 (cid:18) − i (cid:19) + X i =0 (cid:18) − i (cid:19) = 1 + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) = 5 f (5) = 1 + X k =0 3 − k X i =0 (cid:18) k − k − i (cid:19) = 1 + X i =0 (cid:18) − i (cid:19) + X i =0 (cid:18) − i (cid:19) + X i =0 (cid:18) − i (cid:19) + X i =0 (cid:18) − i (cid:19) = 1 + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) + (cid:18) (cid:19) = 8 3. Conclusion
In this paper, we give a closed-form representation for Fibonacci numbers using Fibonaccitrees. A similar approach can be attempted for finding a closed-form representation for Lucasand Bernoulli numbers. 4.
ReferencesReferences [1] G.M. Adelson-Velskii and E.M. Landis, An algorithm for the organization of information, Soviet Math.Dokl., 3 (1962) 1259-1262.[2] R.M. Capocelli, A Note on Fibonacci Trees and the Zeckendorf Representation of Integers, The FibonacciQuaterly, 26, 4 (1988) 318-324.[3] Eric W., “Binet’s Fibonacci Number Formula” from MathWorld.[4] Y. Horibe, Notes on Fibonacci Trees and Their Optimality, The Fibonacci Quaterly, 21, 2 (1983) 118-128.[5] N. Jia, K.W. Mclaughlin, Fibonacci trees: A study of the asymptotic behavior of Balaban’s index, Com-munications in Mathematical and in Computer Chemistry (MATCH), 51 (2004) 79-95.[6] S.G. Wagner, The Fibonacci number of Fibonacci trees and a related family of polynomial recurrencesystems, Fibonacci Quart. 45, 3 (2007) 247253.
School of Information Technology and Engineering, VIT University, Vellore, India
E-mail address : [email protected]@vit.ac.in