CALCULATION OF FIBONACCI VECTORS

CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: ovak@ithaca.edu Draft of 2008 May 19 ABSTRACT 1

1. INTRODUCTION This chapter shows how to calculate ay Fiboacci vector directly, orecursively, geeralizig Biet s formula for the origial Fiboacci sequece. Let be the dimesio of the vector space. 2. SECTION 2 Let F represet the sequece of Fiboacci vectors of dimesio. Let F k represet the Fiboacci vector of idex k, the vector that is displaced by k steps i the sequece from the seed vector F 0. To move oe step from a give vector F k to the ext vector F k + 1 i the sequece is accomplished by applyig the -dimesioal Fiboacci vector sequece augmetatio matrix Q. F T k + 1 Q F T k 1 The superscript T attached to the symbol of a vector or matrix idicates the traspose of that etity. Because we usually defie vectors without such a superscript to represet row vectors, for example F k, the symbol of a vector with a T superscript represets a colum vector, for example F T k. I this recursive defiitio of a Fiboacci vector sequece, the augmetatio matrix Q trasforms the give vector ito the ext vector. 0 0... 0 1 0 0... 1 1 Q.... 2 0 1... 1 1 1 1... 1 1 The elemets of the augmetatio matrix are represeted by Q i, j, where 1 i is the row idex ad 1 j is the colum idex. 0 i + j Q i, j 1 i + j > The effect or the applicatio of the augmetatio matrix is to make the ith compoet of the ext vector equal to the sum of the last i + 1 compoets of the give vector. Note that, sice the augmetatio matrix is a symmetric matrix, Q T Q, we ca write equatio 1 without superscript T s. 3 F k + 1 F k Q 4 The recursive defiitio of the sequece is ot eough, however. We also eed to specify a startig poit, the seed vector of the sequece. For a Fiboacci vector sequece, we specify the followig seed vector F 0. F 0 0 0 0 1 5 2

All but the last compoet of the seed vector are zeros; the last compoet is a oe. 0 1 j < F 0, j 1 j 6 Note that we could have specified a differet seed vector, either trivially by specifyig a differet vector i the same sequece, or sigificatly by specifyig a vector ot i this sequece ad geeratig a o-fiboacci vector sequece. But we shall restrict our attetio here to Fiboacci sequeces grow from the specified seed vectors. The set of the first Fiboacci vectors F k for 0 k 1 forms the the seed matrix F 0 1. F 0 F 0 F 1 1. F 1 F 0 F 0 Q. F 0 Q 1 The kth row of the seed matrix is the Fiboacci vector of idex k 1, F k 1. 7 F 0 1 k F k 1 F 0 Q k 1 8 The characteristic polyomial of degree i the variable x is D x, D x Q xi 9 where the pair of vertical lies extracts the determiat of the eclosed matrix ad I is the by idetity matrix. It may be expressed explicitly as a polyomial, a power series of degree i the variable x. D x d jx j 10 j0 I aother chapter, we determie the values of the coefficiets d j. This polyomial has roots x i for 1 i. The polyomial may also be writte i terms of these roots. D x 1 i1 x x i 11 The roots of the characteristic polyomial are real ad distict, so that they may be ordered. We will place the roots i ascedig order, so that i < j x i < x j. The ordered set of roots is the root vector x. x x 1 x 2 x It is coveiet to have a otatio for a vector, each compoet of which is a certai power of the same compoet of aother vector. Therefore, defie the jth power of a vector, v j, as the vector the ith compoet of which is the jth power of the ith compoet of the vector v. 12 v j i vi j 13 3

Thus, for example, we will write x j i for the ith compoet of the jth power of the vector x, istead of its equivalet expressio x i j for the jth power of the ith compoet of the vector x. The defiig property of the roots of the characteristic polyomial is that, if the variable x is set equal to the value of oe of its roots x i, the characteristic polyomial evaluates to zero. D x i d jx j i 0 14 j0 Let us form arbitrary liear combiatios A k of the kth powers of the roots x i of the characteristic polyomial. A k x k a x k ia i 15 Thik of this as the ier scalar or dot product of x k, the kth power of the -dimesioal root vector x, with a arbitrary -dimesioal colum vector a. Such arbitrary combiatios satisfy the followig relatio. i1 d ja j 0 16 j0 The proof of this relatio 16 follows straightforwardly from the defiitio of the arbitrary combiatios 15 ad the defiig property of the roots of the characteristic polyomial 14 by rearragemet. d ja j j0 d j x j ia i j0 i1 a i d jx j i i0 j0 a id x i 0 17 Now, let us form special liear combiatios of the roots of the characteristic polyomial to represet the compoets of Fiboacci vectors. Form the jth compoet of the kth Fiboacci vector, F k, j as the ier product of x k with a differet -dimesioal colum vector f, j for each compoet. F k, j x k f, j i0 x k if i, j 18 The symbol f represets a matrix, the Fiboacci root power combiatio matrix; f i, j, the elemet i its ith row ad jth colum; f, j, its jth colum vector; ad f i,, its ith row vector. The kth Fiboacci vector is represeted by F k. F k x k f i1 x k if i, 19 We do ot kow the value of the combiatio matrix f, but if we could figure it out, the we would have a procedure for calculatig ay Fiboacci vector, without havig to calculate all the vectors i betwee. i1 Let us represet aew the seed matrix F 0 1, which we defied i 7, by substitutig ito each of its rows this represetatio 19 of each of the first Fiboacci vectors as combiatios of powers of the root 4

vector. F 0 F 0 F 1 1. F 1 x 0 f x 1 f. x 1 f 1 x. x 1 Here, the symbol 1 represets the -dimesioal row vector, all of whose elemets are 1 s. f 20 Defie the root matrix X as the matrix whose ith row, X i,, is x i 1, the i 1th power of the root vector x. X i, x i 1 21 Thus, the elemet X i, j of the root matrix X i its ith row ad jth colum is the i 1th power of the jth root of the characteristic polyomial of degree. X i, j x i 1 j 22 Displayed i more detail, the root matrix X appears as follows. x 0 1 1 1... 1 x 1 x x 1 x 2... x X..... x 1 1 x 1 2... x 1 x 1 Where have you see that matrix before? x 1 23 I equatio 20, you saw that the seed matrix is the product of what you ow recogize as the root matrix X ad the combiatio matrix f. F 0 1 X f 24 We ca fid the combiatio matrix f by multiplyig this equatio by the iverse of the root matrix X 1. f X 1 F 0 1 Now, substitute this ito equatio 19 to fid the kth Fiboacci vector. 25 Fially, we fid that the kth Fiboacci vector is the product of the kth power of the root vector x k, the iverse of the root matrix X 1 ad the seed matrix F 0 1, i that order. F k x k X 1 F 0 1 What we have foud here is a procedure for calculatig ay Fiboacci vector without havig to calculate all the vectors i betwee. To costruct the seed matrix, you do have to calculate the first 1 vectors the seed vector is give. Ad to costruct the root vector ad the root matrix, you do have to determie the roots 26 5

x i for 1 i of the characteristic polyomial D x. By cosiderig the followig example, you will see that what we have here i equatio 26 is the geeralizatio to Fiboacci vectors of Biet s formula for calculatig the origial Fiboacci umbers, which are the compoets of the vectors of the two-dimesioal Fiboacci vector sequece F 2. Cosider the case 2. The characteristic polyomial is D 2 x 1 x + x 2. The root vector x 2 is x 2 1, x 2 2, but let us write it as x 1, x 2 for brevity i this case. x 1, x 2 1 φ, φ 1 5, 2 1 + 5 2 1 2 cos α 2, 2 cos α 2 27 where α 2 π 5 is the miimal agle betwee edges sides or diagoals of a regular petago. Now, let us apply the geeral formula 26 for calculatig Fiboacci vectors i dimesios to calculate the kth Fiboacci vector i 2 two dimesios. 1 F 2 k x k 1, x k 1 1 0 1 2 1 φ k, φ k 1 1 x 1 x 2 1 1 1 φ φ φ k 1 φ k, φ k 1 1 φ k 1 + φ k 1 φ k 0 1 1 1 1 F k, F k 1 + F k 2φ 1 F k, F k + 1 28 where F k xk 2 x k 1 φk 1 φ k x 2 x 1 2φ 1 1+ k 5 2 5 1 k 5 2 2 cos α 2 k 1 2 cos α 2 k 4 cos α 2 1 29 Observe that this is Biet s formula for calculatig the kth Fiboacci umber. Thus, we see that the geeral formula 26 for calculatig Fiboacci vectors i dimesios is the geeralizatio of Biet s formula from 2 two to > 2 higher dimesioal Fiboacci vectors. XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX This chapter will ed here. The followig material o the characteristic polyomial equatio ad its coefficiets will move to aother chapter. 2. FIBONACCI CHARACTERISTIC POLYNOMIALS Juge ad Hoggatt [2] derive formulas for the Fiboacci characteristic polyomials D x. For eve dimesios 2m, the characteristic polyomials are give by their equatio 21. D 2m x [m/2] k0 1 k m k k k0 2x 2 1 m 2k x 4k [m 1/2] m k 1 1 + xx 1 k 2x 2 1 m 2k 1 x 4k 30 k 6

For odd dimesios 2m + 1, the polyomials are give by their equatio 22, which, whe corrected, reads as follows. [m/2] m k D 2m+1 x 1 x 1 k 2x 2 1 m 2k x 4k k k0 [m 1/2] +x 3 k0 m k 1 1 k 2x 2 1 m 2k 1 x 4k 31 k We shall say that these formulas express the polyomials implicitly, ot explicitly, as a power series i x, which we will do later. Note that the latter equatio is derived from Juge ad Hoggatt s equatio 14, which, whe corrected, reads as follows. 1 x + x 3 t 1 2x 2 1t + x 4 t 2 1 D 2m+1 xt 32 The characteristic polyomials satisfy the followig recursio relatio. 0 D +4 x 2x 2 1D +2 x x 4 D x 33 Followig are the first several characteristic polyomials, D x, where is the degree of the polyomial. D 0 x +1 D 1 x +1 x D 2 x 1 x + x 2 D 3 x 1 + x + 2x 2 x 3 D 4 x +1 + x 3x 2 2x 3 + x 4 D 5 x +1 x 4x 2 + 3x 3 + 3x 4 x 5 D 6 x 1 x + 5x 2 + 4x 3 6x 4 3x 5 + x 6 D 7 x 1 + x + 6x 2 5x 3 10x 4 + 6x 5 + 4x 6 x 7 D 8 x +1 + x 7x 2 6x 3 + 15x 4 + 10x 5 10x 6 4x 7 + x 8 34 The characteristic polyomials ca be expressed as a power series with coefficiets D,j, where is the degree of the polyomial ad j is the degree of the term. D x D,j x j 35 The followig is a Table of the coefficiets of the characteristic polyomials, D,j. j0 7

j 0 1 0 +1 2 1 +1 1 3 2 1 1 +1 4 3 1 +1 +2 1 5 4 +1 +1 3 2 +1 6 5 +1 1 4 +3 +3 1 7 6 1 1 +5 +4 6 3 +1 8 7 1 +1 +6 5 10 +6 +4 1 8 +1 +1 7 6 +15 +10 10 4 +1 Table 1: Fiboacci Characteristic Polyomial Coefficiets, D,j Rows represet the coefficiets of the polyomials of degree 8. Right leaig colums represet the coefficiets of the terms of degree j, where 0 j. This Table is remiiscet of Pascal s triagular table of the biomial coefficiets. We shall call it Fiboacci s triagle ad the coefficiets, Fiboacci coefficiets. While the biomial coefficiets are all positive, the Fiboacci coefficiets show a iterestig patter of alteratig positive ad egative sigs. The magitudes of the Fiboacci coefficiets of fixed j vary with icreasig just as the biomial coefficiets do. But while Pascal s triagle is symmetric about the midlie j /2, the Fiboacci triagle is defiitely ot symmetric. The Fiboacci coefficiets are geerated by the followig formula. j + [j/2] D,j 1 j+[ j/2] j Here, the square brackets [ ] idicate the floor or roud-dow fuctio, which yields the largest iteger smaller tha the argumet that the brackets cotai. The characteristic polyomial may ow be writte i terms of the Fiboacci coefficiets. D x 36 j + [j/2] 1 j+[ j/2] x j 37 j j0 This form of the characteristic polyomial is much simpler ad easier to use ad uderstad tha the implicit form of the polyomial derived by Juge ad Hoggatt [2]. It is idetical to explicit form of the polyomials derived by Raey [3] i his Theorem 9. 8

Let us fid the recursio relatios obeyed by the Fiboacci characteristic polyomial coefficiets. Substitute the defiitio of the polyomial as a power series, equatio 35, ito the recursio relatio for the polyomials, equatio 33, ad equate terms of the same degree i x. Thus, we fid the followig set of relatios amog the polyomial coefficiets. D +2,0 D,0 D +2,1 D,1 D +2,2 2D,0 D,2 D +2,3 2D,1 D,3 D +4,j+4 2D +2,j+2 D +2,j+4 D,j 38 You may cofirm that the polyomial coefficiets, derived from the formula for them, equatio 36, ad displayed i Table 1, satisfy all these relatios. Now that we have foud the Fiboacci characteristic polyomial coefficiets, let us make some use of them. Earlier, we displayed the recursio relatio, equatio, that defies Fiboacci vector sequeces. From a give vector, it says how to calculate, compoet by compoet, the ext vector i the sequece. Let us ow cosider a related problem. How ca you calculate a particular compoet of the ext vector i the sequece, if you kow, ot the compoets of the oe precedig vector, but the same sigle compoet of the precedig vectors? To fid the solutio to this problem, let us briefly retur to the origial Fiboacci sequece. This sequece is defied, as you well kow, by a recursive relatio: ay umber i the sequece is the sum of the two precedig umbers. F k + 2 F k + 1 + F k 39 Now, traslate this familiar equatio ito the symbols of Fiboacci vectors ad rearrage a bit. F 2 k, i F 2 k + 1, i + F 2 k + 2, i 0 40 The sigs of the three terms are just the three coefficiets D 2,j of the characteristic polyomial of degree 2. 2 D 2,0 F 2 k, i + D 2,1 F 2 k + 1, i + D 2,2 F 2 k + 2, i D 2,j F k + j, i 0 41 Now, geeralize. Fiboacci vector compoets are recursively related by the followig geeralizatio of the origial Fiboacci recursive relatio. D,j F k + j, i 0 42 j0 We postpoe proof that this relatio amog Fiboacci vector compoets follows from the recursive relatio betwee cosecutive Fiboacci vectors, equatio, which defies Fiboacci vector sequeces. j0 9

Now, let s put this geeral priciple ito a form useful for calculatio. Isolate the term of highest degree, F k +, i, the coefficiet of which is D, 1, ad we have the aswer to our problem. 1 F k +, i 1 +1 D,j F k + j, i 0 43 j i+1 j1 j0 The ith compoet of the k +th vector i the sequece, F k +, i, equals the sum from j 0 to j 1 of the products of the ith compoet of the k + jth vector ad the polyomial coefficiet D,j. You ca compute ay compoet of a Fiboacci vector, if you kow the same compoet of the precedig vectors: you multiply each such compoet by the appropriate polyomial coefficiet ad add these products to obtai the result. The golde sum: For a give dimesio ad idex i, 1 i, the sum of the greatest i golde umbers, from r i + 1 to r, is the product of the ith golde umber r i ad the greatest oe golde umber r. i r j r i + j r r i 44 Proof of this rule follows simply from equatio 18 of our earlier paper [1], the golde system of equatios, which may be writte as follows. r j r r j + 1 r j 45 Sum both sides from j 1 to j i 1. i 1 r j j1 1 j i+1 Because r 1 1, the golde sum follows. i 1 r j r r j + 1 r j r r i r 1 46 j1 6. CONCLUSION 10

REFERENCES Refereces [1] Stuart D. Aderso ad Dai Novak, Fiboacci Vector Sequeces ad Regular Polygos, Fiboacci Quarterly,??:?,???-??? 20??. [2] Bjare Juge ad V. E. Hoggatt, Jr., Polyomials Arisig from Reflectios across Multiple Plates, Fiboacci Quarterly, 11:3, 285-291 1973. [3] George N. Raey, Geeralizatio of the Fiboacci Sequece to Dimesios, Ca. J. Math., 18, 332-349 1966. AMS Classificatio Number: 11B39 11