A Simplified Binet Formula for k-generalized Fibonacci Numbers

Transcription

1 A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 Zhaohui Du Shaghai, Chia Abstract I this paper, we preset a Biet-style formula that ca be used to produce the k-geeralized Fiboacci umbers that is, the Triboaccis, Tetraaccis, etc.). Furthermore, we show that i fact oe eeds oly take the iteger closest to the first term of this Biet-style formula i order to geerate the desired sequece. 1 Itroductio Let k ad defie F k), the th k-geeralized Fiboacci umber, as follows: 0, if < 1; 1, if = 1; F k) 1 + F k) + + F k) k, if > 1 F k) = These umbers are also called geeralized Fiboacci umbers of order k, Fiboacci k- step umbers, Fiboacci k-sequeces, or k-boacci umbers. Note that for k =, we have F ) = F, our familiar Fiboacci umbers. For k = 3 we have the so-called Triboaccis sequece umber A i Sloae s Ecyclopedia of Iteger Sequeces), followed by the Tetraaccis A000078) for k = 4, ad so o. Accordig to Kessler ad Schiff [6], these umbers also appear i probability theory ad i certai sortig algorithms. We preset here a chart of these umbers for the first few values of k: k ame first few o-zero terms Fiboacci 1, 1,, 3,, 8, 13, 1, 34,... 3 Triboacci 1, 1,, 4, 7, 13, 4, 44, 81,... 4 Tetraacci 1, 1,, 4, 8, 1, 9, 6, 108,... Petaacci 1, 1,, 4, 8, 16, 31, 61, 10,... 1

2 We remid the reader of the famous Biet formula also kow as the de Moivre formula) that ca be used to calculate F, the Fiboacci umbers: [ ) F = 1 ) ] = α β α β for α > β the two roots of x x 1 = 0. For our purposes, it is coveiet ad ot particularly difficult) to rewrite this formula as follows: F = α 1 β 1 + 3α ) α β ) β 1 1) We leave the details to the reader. Our first ad very mior) result is the followig represetatio of F k) : Theorem 1. For F k) the th k-geeralized Fiboacci umber, the F k) = k i=1 for α 1,..., α k the roots of x k x k 1 1 = 0. α i 1 + k + 1)α i ) α 1 i ) This is a ew presetatio, but hardly a ew result. There are may other ways of represetig these k-geeralized Fiboacci umbers, as see i the articles [, 3, 4,, 7, 8, 9]. Our Eq. ) of Theorem 1 is perhaps slightly easier to uderstad, ad it also allows us to do some aalysis as see below). We poit out that for k =, Eq. ) reduces to the variat of the Biet formula for the stadard Fiboacci umbers) from Eq. 1). As show i three distict proofs [9, 10, 13], the equatio x k x k 1 1 = 0 from Theorem 1 has just oe root α such that α > 1, ad the other roots are strictly iside the uit circle. We ca coclude that the cotributio of the other roots i Eq. will quickly become trivial, ad thus: F k) α 1 + k + 1)α ) α 1 for sufficietly large. 3) It s well kow that for the Fiboacci sequece F ) Eq. 3) is = 0, as show here: = F, the sufficietly large i F ) error

3 It is perhaps surprisig to discover that a similar statemet holds for all the k-geeralized Fiboacci umbers. Let s first defie rdx) to be the the value of x rouded to the earest iteger: rdx) = x + 1. The, our mai result is the followig: Theorem. For F k) the th k-geeralized Fiboacci umber, the ) F k) α 1 = rd + k + 1)α ) α 1 for all k ad for α the uique positive root of x k x k 1 1 = 0. We poit out that this theorem is ot as trivial as oe might thik. Note the error term for the geeralized Fiboacci umbers of order k = 6, as see i the followig chart; it is ot mootoe decreasig i absolute value F 6) α 1 +7α ) α error We also poit out that ot every recurrece sequece admits such a simple formula as see i Theorem. Cosider, for example, the scaled Fiboacci sequece 10, 10, 0, 30, 0, 80,..., which has Biet formula: 10 This ca be writte as rd 1 + ) ). ) ), but oly for. As aother example, the sequece 1,, 8, 4, 80,... defied by G = G 1 + 4G ) ca be writte as G = 1 + ) 1 ), but because both 1 + ad 1 have absolute value greater tha 1, the it would be impossible to express G i terms of just oe of these two umbers. Previous Results We poit out that for k = 3 the Triboacci umbers), our Theorem was foud earlier by Spickerma [11]. His formula modified slightly to match our otatio) reads as follows, where α is the real root, ad σ ad σ are the two complex roots, of x 3 x x 1 = 0: F 3) = rd α α σ)α σ) α 1 3 ) 4)

4 It is ot hard to show that for k = 3, our coefficiet Spickerma s coefficiet α α σ)α σ) α 1 +k+1)α ) from Theorem is equal to. We leave the details to the reader. I a subsequet article [1], Spickerma ad Joyer developed a more complex versio of our Theorem 1 to represet the geeralized Fiboacci umbers. Usig our otatio, ad with {α i } the set of roots of x k x k 1 1 = 0, their formula reads F k) = k i=1 α k i α k+1 i αi k i ) k + 1)α 1 It is surprisig that eve after calculatig out the appropriate costats i their Eq. ) for k 10, either Spickerma or Joyer oted that they could have simply take the first term i Eq. ) for all 0, as Spickerma did i Eq. 4) for k = 3. The Spickerma-Joyer Eq. ) was exteded by Wolfram [13] to the case with arbitrary startig coditios rather tha the iitial sequece 0, 0,..., 0, 1). I the ext sectio we will show that our Eq. ) i Theorem 1 is equivalet to the Spickerma-Joyer formula give above ad thus is a special case of Wolfram s formula). Fially, we ote that the polyomials x k x k 1 1 i Theorem 1 have bee studied rather extesively. They are irreducible polyomials with just oe zero outside the uit circle. That sigle zero is located betwee 1 k ) ad as see i Wolfram s article [13]; Miles [9] gave earlier ad less precise results). It is also kow [13, Lemma 3.11] that the polyomials have Galois group S k for k 11; i particular, their zeros ca ot be expressed i radicals for k 11. Wolfram cojectured that the Galois group is always S k. Cipu ad Luca [1] were able to show that the Galois group is ot cotaied i the alteratig group A k, ad for k 3 it is ot -ilpotet. They poit out that this meas the zeros of the polyomials x k x k 1 1 for k 3 ca ot be costructed by ruler ad compass, but the questio of whether they are expressible usig radicals remais ope for k 1. 3 Prelimiary Lemmas First, a few statemets about the the umber α. Lemma 3. Let α > 1 be the real positive root of x k x k 1 x 1 = 0. The, I additio, 1 k < α < 6) 1 < α < for k 4. 7) Proof. We begi by computig the followig chart for k : k 1 k α

5 It s clear that 1 1 < α < for k ad that < α < for 4 k. We ow k focus o k 6. At this poit, we could fiish the proof by appealig to 1 k ) < α < as see i the article [13, Lemma 3.6], but here we preset a simpler proof. Let fx) = x 1)x k x k 1 x 1) = x k+1 x k + 1. We kow from our earlier discussio that fx) has oe real zero α > 1. Writig fx) as x k x ) + 1, we have f 1 ) = 1 ) k ) ) For k 6, it s easy to show < ) k = 1 k < 3 3) 1 ) k Substitutig this iequality ito the right-had side of 8), we ca re-write 8) as f 1 ) ) 1 < ) + 1 = 0. Fially, we ote that f) = k+1 k + 1 = 1 > 0, so we ca coclude that our root α is withi the desired bouds of 1/ ad for k 6. We ow have a lemma about the coefficiets of α 1 i Theorems 1 ad. Lemma 4. Let k be a iteger, ad let m k) x 1 x) = + k + 1)x ). The, 1. m k) 1/k) = 1.. m k) ) = m k) x) is cotiuous ad decreasig o the iterval [ 1/k, ). 4. m k) x) > 1 x o the iterval 1/k, ). Proof. Parts 1 ad are immediate. As for 3, ote that we ca rewrite m k) x) as ) m k) x) = 1 1 k k + 1 x ) k+1 which is simply a scaled traslatio of the map y = 1/x. I particular, sice this m k) x) has a vertical asymptote at x =, the by parts 1 ad we ca coclude that k+1 mk) x) is ideed cotiuous ad decreasig o the desired iterval. To show part 4, we first ote that i solvig 1 = x mk) x), we obtai a quadratic equatio with the two itersectio poits x = ad x = k. It s easy to show that 1 < x mk) x) at x = 1/k, ad sice both fuctios 1 ad x mk) x) are cotiuous o the iterval [ 1/k, ) ad itersect oly at x = ad x = k, we ca coclude that 1 < x mk) x) o the desired iterval.

6 Lemma. For a fixed value of k ad for k, defie E to be the error i our Biet approximatio of Theorem, as follows: E = F k) α 1 + k + 1)α ) α 1 = F k) m k) α) α 1, for α the positive real root of x k x k 1 x 1 = 0 ad m k) as defied i Lemma 4. The, E satisfies the same recurrece relatio as F k) : E = E 1 + E + + E k for ). Proof. By defiitio, we kow that F k) satisfies the recurrece relatio: F k) = F k) F k) k 9) As for the term m k) α) α 1, ote that α is a root of x k x k 1 1 = 0, which meas that α k = α k , which implies m k) α) α 1 = m k) α)α + + m k) α)α k+1) 10) We combie Equatios 9) ad 10) to obtai the desired result. 4 Proof of Theorem 1 As metioed above, Spickerma ad Joyer [1] proved the followig formula for the k- geeralized Fiboacci umbers: F k) = k i=1 α k i α k+1 i αi k i 11) k + 1)α 1 Recall that the set {α i } is the set of roots of x k x k 1 1 = 0. We ow show that this formula is equivalet to our Eq. ) i Theorem 1: F k) = k i=1 α i 1 + k + 1)α i ) α 1 i 1) Sice αi k α k 1 i 1 = 0, we ca multiply by α i 1 to get α k+1 i αi k = 1, which implies α i ) = 1 α k i. We use this last equatio to trasform 1) as follows: α i 1 + k + 1)α i ) = α i 1 + k + 1) α k i ) = α αi k k+1 i αi k k + 1) This establishes the equivalece of the two formulas 11) ad 1), as desired. 6

7 Proof of Theorem Let E be as defied i Lemma. We wish to show that E < 1 for all k. We proceed by first showig that E < 1 for = 0, the for = 1,, 3,..., k, the for = 1, ad fially that this implies E < 1 for all k. To begi, we ote that sice our iitial coditios give us that F k) = 0 for = 0, 1,,..., k, the we eed oly show m k) α) α 1 < 1/ for those values of. Startig with = 0, it s easy to check by had that m k) α) α 1 < 1/ for k = ad 3, ad as for k 4, we have the followig iequality from Lemma 3: which implies Also, by Lemma 4, so thus: 1 < α, α 1 < 6k 1. m k) α) < m k) 1/) = 1 k 1, m k) α) α 1 < 1 k 1 6k 1 < ) 1 k 1) < 1, as desired. Thus, 0 < m k) α) α 1 < 1/ for all k, as desired. Sice α 1 < 1, we ca coclude that for = 1,,..., k, the E = m k) α) α 1 < 1/. Turig our attetio ow to E 1, we ote that F k) 1 = 1 agai by defiitio of our iitial coditios) ad that 1 = m) < mα) < m 1/k) = 1 which immediately gives us E 1 < 1/. As for E with, we kow from Lemma that E = E 1 + E + + E k for ) Suppose for some that E 1/. Let 0 be the smallest positive such. Now, subtractig the followig two equatios: gives us: E 0 +1 = E 0 + E E 0 k 1) E 0 = E E E 0 k E 0 +1 = E 0 E 0 k Sice E 0 E 0 k the first, by assumptio, beig larger tha, ad the secod smaller tha, 1/), we ca coclude that E 0 +1 > E 0. I fact, we ca apply this argumet repeatedly to show that E 0 +i > > E 0 +1 > E 0. However, this cotradicts the observatio from Eq. 3) that the error must evetually go to 0. We coclude that E < 1/ for all, ad thus for all k. 7

8 6 Ackowledgemet The first author would like to thak J. Siehler for ispirig this paper with his work o Triboacci umbers. Refereces [1] M. Cipu ad F. Luca, O the Galois group of the geeralized Fiboacci polyomial, A. Ştiiţ. Uiv. Ovidius Costaţa Ser. Mat ), [] David E. Ferguso, A expressio for geeralized Fiboacci umbers, Fiboacci Quart ), [3] I. Flores, Direct calculatio of k-geeralized Fiboacci umbers, Fiboacci Quart. 1967), [4] Hyma Gabai, Geeralized Fiboacci k-sequeces, Fiboacci Quart ), [] Da Kalma, Geeralized Fiboacci umbers by matrix methods, Fiboacci Quart ), [6] David Kessler ad Jeremy Schiff, A combiatoric proof ad geeralizatio of Ferguso s formula for k-geeralized Fiboacci umbers, Fiboacci Quart ), [7] Gwag-Yeo Lee, Sag-Gu Lee, Ji-Soo Kim, ad Hag-Kyu Shi, The Biet formula ad represetatios of k-geeralized Fiboacci umbers, Fiboacci Quart ), [8] Claude Levesque, O mth order liear recurreces, Fiboacci Quart ), [9] E. P. Miles, Jr., Geeralized Fiboacci umbers ad associated matrices, Amer. Math. Mothly ), [10] M. D. Miller, Mathematical Notes: O Geeralized Fiboacci Numbers, Amer. Math. Mothly ), [11] W. R. Spickerma, Biet s formula for the Triboacci sequece, Fiboacci Quart ), [1] W. R. Spickerma ad R. N. Joyer, Biet s formula for the recursive sequece of order k, Fiboacci Quart. 1984), [13] D. A. Wolfram, Solvig geeralized Fiboacci recurreces, Fiboacci Quart ),

9 000 Mathematics Subject Classificatio: Primary 11B39, Secodary 11C08, 33F0, 6D0. Keywords: k-geeralized Fiboacci umbers, Biet, Triboacci, Tetraacci, Petaacci. Cocered with sequeces A000073, A000078, ad A00191.) Received February 3,