PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES

Transcription

1 appeared n: Fbonacc Quarterly 38(2000), pp PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES Arthur T. Benjamn Dept. of Mathematcs, Harvey Mudd College, Claremont, CA Jennfer J. Qunn Dept. of Mathematcs, Occdental College, Los Angeles, CA Francs Edward Su Dept. of Mathematcs, Harvey Mudd College, Claremont, CA Introducton Fbonacc numbers arse n the soluton of many combnatoral problems. They count the number of bnary sequences wth no consecutve zeros, the number of sequences of 1 s and 2 s whch sum to a gven number, and the number of ndependent sets of a path graph. Smlar nterpretatons exst for Lucas numbers. Usng these nterpretatons, t s possble to provde combnatoral proofs whch shed lght on many nterestng Fbonacc and Lucas denttes [1, 3]. In ths paper, we extend the combnatoral approach to understand relatonshps among generalzed Fbonacc numbers. Gven G 0 and G 1, a generalzed Fbonacc sequence G 0, G 1, G 2,... s defned recursvely by G n = G n 1 + G n 2 for n 2. Two mportant specal cases are the classcal Fbonacc sequence F n (F 0 = 0 and F 1 = 1) and the Lucas sequence L n (L 0 = 2 and L 1 = 1). These sequences satsfy numerous relatonshps. Many are documented n Vajda [6], where they are proved by algebrac means. Our goal s to recount these denttes by combnatoral means. We ntroduce several combnatoral technques whch allow us to provde new proofs of nearly all the denttes n Vajda [6] nvolvng generalzed Fbonacc numbers. We show that n the framework of phased tlngs, these denttes follow naturally as the tlngs are counted, represented, and transformed n clever ways. These technques are developed n the next several sectons. In the fnal secton, we dscuss possble extensons. 2. Combnatoral Interpretaton Recall that F n+1 counts the number of sequences of 1 s and 2 s whch sum to n. Equvalently, F n+1 counts the number of ways to tle a 1 n rectangle (called an n-board consstng of cells labelled 1,..., n) wth 1 1 squares and 1 2 domnoes. For combnatoral convenence, we defne f n = F n+1. Then f n s the number of ways to tle an n-board wth squares and domnoes. When G 0 and G 1 are non-negatve ntegers, we shall obtan an analogous combnatoral nterpretaton of the generalzed Fbonacc numbers G n. Defne a phased n-tlng to be a tlng of an n-board by squares and domnoes n whch the last tle s dstngushed n a certan way. Specfcally, f the last tle s a domno, t can be assgned one of G 0 possble phases, and f the last tle s a square t can be assgned one of G 1 possble phases. For example, when G 0 = 5 and G 1 = 17, there are G 3 = 39 phased tlngs of length 3 as follows: There are 5 of the form (square, phased domno); 17 of the form (domno, phased square); and 17 of the form (square, square, phased 1

2 2 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES square). In general, let g 0 = G 0, g 1 = G 1, and for n 2, let g n count the number of phased n-tlngs. By condtonng on whether the frst tle s a square or domno we obtan the dentty g n = g n 1 + g n 2 for n 2. Hence g n = G n, gvng the desred nterpretaton. Ths combnatoral defnton can be extended to n = 1 and n = 0. Clearly G 1 counts the number of phased 1-tlngs. It wll be convenent to assgn the last tle of a 0-board one of the G 0 domno phases. Notce when there exsts only one domno phase and only one square phase, we recover our orgnal nterpretaton of f n. Prevous nterpretatons of the Lucas numbers L n [1, 4, 5] counted the number of ways to tle a crcular n-board by squares or domnoes. Snce L 0 = 2 and L 1 = 1, a phased n-board tlng can end wth a phase one domno, a phase two domno, or a phase one square. In all three cases, the correspondng crcular n-board tlng arses by frst glung cells n and 1 together. Tlngs that end n a phase two domno are then rotated one cell to obtan a crcular tlng wth a domno coverng cells n and Elementary Identtes Before launchng nto more sophstcated technques, we demonstrate how our combnatoral nterpretaton of G n yelds quck proofs of some basc denttes. For nstance, by condtonng on whether the last tle s a phased domno or a phased square, we mmedately obtan for n 2, G n = G 0 f n 2 + G 1 f n 1. More denttes are obtaned by condtonng on other events. Consder Identty 1 (Vajda (33)). G k = G n+2 G 1. The rght hand sde of ths equalty counts all phased (n+2)-tlngs contanng at least one domno (there are G 1 phased tlngs consstng of all squares). The left hand sde s obtaned by condtonng on the poston of the frst domno. If the frst domno covers cells n k+1 and n k+2 (0 k n), then the precedng cells are covered by squares and the remanng cells can be covered G k ways. Smlarly there are G 2n G 0 phased 2n-tlngs wth at least one square. By condtonng on the poston of the frst square we obtan Identty 2 (Vajda (34)). G 2k 1 = G 2n G 0. A phased (2n + 1)-tlng must contan a frst square, whch leads to Identty 3 (Vajda (35)). k=1 G 1 + G 2k = G 2n+1. k=1 The G 1 term on the left hand sde counts those boards that begn wth n domnoes followed by a phased square. To prove Identty 4 (Vajda (8)). G m+n = f m G n + f m 1 G n 1

3 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 3 we consder whether or not a phased (m + n)-tlng can be separated nto an (unphased) m-tlng followed by a phased n-tlng. There are f m G n tlngs breakable at cell m. The unbreakable tlngs must contan a domno coverng cells m and m + 1; the remanng board can be covered f m 1 G n 1 ways. 4. Bnomal Identtes Vajda contans several denttes nvolvng generalzed Fbonacc numbers and bnomal coeffcents. All of these are specal cases of the followng two denttes: Identty 5 (Vajda (46)). Identty 6 (Vajda (66)). G n+p = G m+(t+1)p = p =0 p =0 ( ) p G n, ( ) p ft f p t 1 G m+. When n p, Identty 5 counts phased (n+p)-tlngs by condtonng on the number of domnoes that appear among the frst p tles. Gven an ntal segment of domnoes and p squares, ( ) p counts the number of ways to select the postons for the domnoes among the frst p tles. G n counts the number of ways the remanng n cells can be gven a phased tlng. Identty 6 can be seen as tryng to break a phased (m+(t+1)p)-tlng nto p unphased segments of length t followed by a phased remander. The frst segment conssts of the tles coverng cells 1 through j 1 where j 1 = t f the tlng s breakable at cell t and j 1 = t + 1 otherwse. The next segment conssts of the tles coverng cells j through j 1 + j 2 where j 2 = t f the tlng s breakable at cell j 1 + t and j 2 = t + 1 otherwse. Contnung n ths fashon we decompose our phased tlng nto p tled segments of length t or t + 1 followed by a phased remander of length at least m. Snce the length t + 1 segments must end wth a domno, the term ( ) p f t f p t 1 G m+ counts the number of phased (m + (t + 1)p)-tlngs wth exactly segments of length t. 5. Smultaneous Tlngs Identtes nvolvng squares of generalzed Fbonacc numbers suggest nvestgatng pars of phased tlngs. The rght hand sde of Identty 7 (Vajda (39)). 2 =1 G 1 G = G 2 2n G 2 0 counts ordered pars (A, B) of phased 2n-tlngs where A or B contans at least one square. To nterpret the left hand sde, we defne the parameter k X to be the frst cell of the phased tlng X covered by a square. If X s all domnoes, we set k X equal to nfnty. Snce, n ths case, at least one square exsts n (A, B), the mnmum of k A and k B must be fnte and odd. Let k = mn{k A, k B + 1}. When k s odd, A and B have domnoes coverng cells 1 through k 1 and A has a square coverng cell k. Hence the number of phased pars (A, B) wth odd k s G 2n k G 2n k+1. When k s even, A has domnoes coverng cells 1 through k and B has domnoes coverng cells 1 through k 2 wth a square coverng cell k 1. Hence the number of phased pars (A, B) wth even k s also G 2n k G 2n k+1. Settng = 2n + 1 k gves the desred dentty. Smlarly, the next dentty counts ordered pars of phased (2n + 1)-tlngs that contan an unphased square. Condtonng on the frst unphased square yelds

4 4 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES Identty 8 (Vajda (41)). 2n+1 =2 G 1 G = G 2 2n+1 G 2 1. In the same sprt, our next dentty condtons on the locaton of the frst domno n a par of phased tlngs. Identty 9 (Vajda (43)). G 1 G +2 = G 2 n+1 G 2 1 =1 The rght hand sde counts the number of pars (A, B) of phased (n + 1)-tlngs, where A or B contans at least one domno. Here we defne the parameter l X to be the frst cell of the phased tlng X covered by a domno. If X s all squares, we set l X equal to nfnty. Let l = mn{l A, l B }. The number of phased (n + 1)-tlng pars (A, B) where the lth cell of A s covered by a domno s G n l G n l+2 ; the number of such pars where the lth cell of A s covered by a square s G n l+1 G n l. Ths mples G n l (G n l+2 + G n l+1 ) = G 2 n+1 G 2 1. l=1 Substtutng G n l+3 for G n l+2 + G n l+1 and lettng = n l + 1 yelds the desred dentty. 6. A Transfer Procedure The denttes proved n ths secton all take advantage of the same technque. Before proceedng, we ntroduce helpful notaton. For m 0, defne G m to be the set of all phased m-tlngs wth G 0 domno phases and G 1 square phases. An element A G m created from a sequence of e 1 domnoes, e 2 squares, e 3 domnoes,..., and endng wth a phased tle can be unquely expressed as A = d e1 s e2 d e3 s e4 d et 1 s et p, where p represents the phase of the last tle. All exponents are postve except that e 1 or e t may be 0, and 2e 1 + e 2 + 2e 3 + e e t 1 + e t = m. When e t = 0, the last tle s a domno and p {1,..., G 0 }; when e t 1, the last tle s a square and p {1,..., G 1 }. Lkewse, for n 0, defne H n to be the set of all phased n-tlngs wth H 0 domno phases and H 1 square phases. Notce that the szes of G m and H n are G m and H n respectvely. We ntroduce a transfer procedure T to map an ordered par (A, B) G m H n to an ordered par (A, B ) G m 1 H n+1 where 1 m n. T has the effect of shrnkng the smaller tlng and growng the larger tlng by one unt. For such a par (A, B), defne k = mn{k A, k B }, the frst cell n A or B that s covered by a square. If the kth cell of A s covered by a square and 1 k m 1, then we transfer that square from A to the kth cell of B. Formally, before the transfer we have A = d (k 1)/2 sa and B = d (k 1)/2 b where a G m k and b H n k+1. The transfer yelds A = d (k 1)/2 a and B = d (k 1)/2 sb. If the kth cell of A s covered by a domno and 1 k m 2, then we exchange that domno wth the square n the kth cell of B. Formally, before the exchange, A = d (k+1)/2 a and B = d (k 1)/2 sb where a G m k 1 and b H n k. The exchange yelds A = d (k 1)/2 sa and B = d (k+1)/2 b. We abbrevate ths transformaton by T (A, B) = (A, B ). Notce that our rules do not allow for a phased tle to be transferred or exchanged. Lemma 1. For 1 m n, T establshes an almost one-to-one correspondence between G m H n and G m 1 H n+1. The dfference of ther szes satsfes G m H n G m 1 H n+1 = ( 1) m [G 0 H n m+2 G 1 H n m+1 ].

5 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 5 Proof. Notce when T s defned, T (A, B) has the same k value as (A, B) whch makes T easy to reverse. It remans to enumerate (A, B) G m H n for whch T s undefned and (A, B ) G m 1 H n+1 that do not appear n the mage of T. When m s odd, T s undefned whenever k = m. Here, A = d (m 1)/2 a and B = d (m 1)/2 b where a G 1 and b H n m+1. Hence the doman of T contans G m H n G 1 H n m+1 elements. The elements of G m 1 H n+1 that do not appear n the mage of T have k m and are therefore of the form A = d (m 1)/2 p and B = d (m 1)/2 b where p {1,..., G 0 } and b H n m+2. Hence the mage of T conssts of G m 1 H n+1 G 0 H n m+2 elements. Snce T s one-to-one, we have when m s odd, G m H n G 1 H n m+1 = G m 1 H n+1 G 0 H n m+2. When m s even, T s undefned whenever k m and sometmes undefned when k = m 1. Specfcally, T s undefned when A = d m/2 p and B = d (m 2)/2 b where p {1,..., G 0 } and b H n m+2. Hence the doman of T contans G m H n G 0 H n m+2 elements. The elements of G m 1 H n+1 that do not appear n the mage are of the form A = d (m 2)/2 a and B = d m/2 b where a G 1 and b H n m+1. Hence the mage of T conssts of G m 1 H n+1 G 1 H n m+1 elements. Thus when m s even, we have G m H n G 0 H n m+2 = G m 1 H n+1 G 1 H n m+1. We specalze Lemma 1 by settng m = n and choosng the same ntal condtons for G n and H n to obtan Identty 10 (Vajda (28)). G n+1 G n 1 G 2 n = ( 1) n (G 2 1 G 0 G 2 ). Alternately, settng G m = F m and evaluatng Lemma 1 at m + 1, we obtan Identty 11 (Vajda (9)). For 0 m n, H n m = ( 1) m (F m+1 H n F m H n+1 ). A slghtly dfferent transfer process s used to prove Identty 12 (Vajda (10a)). For 0 m n, G n+m + ( 1) m G n m = L m G n. We construct an almost one-to-one correspondence from L m G n to G m+n, where L m denotes the set of Lucas tlngs of length m. Let (A, B) L m G n. If A ends n a (phase 1) square or a phase 1 domno, then we smply append A to the front of B to create an (m + n)-tlng that s breakable at m. Otherwse, A ends n a phase 2 domno. In ths case, before appendng A to the front of B, we transfer a unt from B to A by a smlar process. (If the frst square occurs n B, then transfer t nto the correspondng cell of A. Otherwse, the frst square of A s exchanged wth the correspondng domno n B.) Ths creates a tlng of G m+n that s unbreakable at m. When m s even, the transfer s undefned for the G n m elements of L m G n where A contans only domnoes, endng wth a phase 2 domno, and B begns wth m/2 domnos. Otherwse the transfer s one-to-one and onto G m+n. When m s odd, the transfer s always defned, but msses the G n m elements of G m+n that begn wth m domnoes. Identty 12 follows. A smlar argument establshes Identty 13 (Vajda (10b)). For 0 m n, G n+m ( 1) m G n m = F m (G n 1 + G n+1 ).

6 6 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES The transfer process T can be refned to allow us to shrnk and grow pars of phased tlngs by more than one unt. Specfcally, we construct an almost one-to-one correspondence between G m H n and G m h H n+h where 1 h m 1 < n. Let F n denote the set of (unphased) n-tlngs. So F n = f n. Gven (A, B) G m H n, defne a transfer process T h as follows: If A s breakable at cell h,.e., A = a 1 a 2 where a 1 F h and a 2 G m h, then we append segment a 1 to the begnnng of B. That s to say, T h (A, B) = (A, B ) where A = a 2 and B = a 1 B. If A s unbreakable at cell h,.e. A = a 1 da 2 where a 1 F h 1 and a 2 G m h 1, then let (A, B ) = T (da 2, B) and (A, B ) = (A, a 1 B ). Notce that B, when defned, wll necessarly begn wth a domno and therefore B wll be unbreakable at h. Dscrepances n T h, mappng G m H n to G m h H n+h, are proportonal (by a factor of f h 1 ) to the dscrepances n T mappng G m h+1 H n to G m h H n+1. Hence for 1 h m 1, Lemma 1 mples (1) G m H n G m h H n+h = ( 1) m h+1 f h 1 (G 0 H n m+h+1 G 1 H n m+h ). Notce that f h 1 H n m+h+1 counts the number of phased (n m + 2h)-tlngs that are breakable at h 1. Hence f h 1 H n m+h+1 = H n m+2h f h 2 H n m+h. Smlarly f h 1 G 1 = G h f h 2 G 0. So equaton (1) can be rewrtten as Rendexng, ths s equvalent to Identty 14 (Vajda (18)). G m H n G m h H n+h = ( 1) m h+1 (G 0 H n m+2h G h H n m+h ). G n+h H n+k G n H n+h+k = ( 1) n (G h H k G 0 H h+k ). Ths dentty s appled, drectly or ndrectly, by Vajda to obtan denttes (19a) through (32). 7. Bnary Sequences There are denttes nvolvng generalzed Fbonacc numbers and powers of 2. Ths leads us to nvestgate the relatonshp between bnary sequences and Fbonacc tlngs. A bnary sequence x = x 1 x 2 x n can be vewed as a set of nstructons for creatng a Fbonacc tlng of length less than or equal to n. Readng x from left to rght, we nterpret 1 s and 01 s as squares and domnoes respectvely. The constructon halts on encounterng a 00 or the end of the sequence. For example, represents the 12-tlng s 3 d 2 sd 2, represents the 9-tlng s 3 d 2 s 2, and represents the 4-tlng ds 2. Bnary sequences that begn wth 00 denote the 0-tlng. Gven n, tlngs of length n and n 1 are represented unquely by bnary sequences of length n that end wth a 1 or 0, respectvely. For k n 2, a k-tlng s represented by 2 n k 2 bnary sequences of length n snce the frst k + 2 bts are determned by the k-tlng followed by 00; the remanng n (k + 2) bts may be chosen freely. Ths yelds the followng dentty: (2) n 2 f n + f n 1 + f k 2 n k 2 = 2 n. By dvdng by 2 n, rendexng, and employng f n+1 = f n + f n 1, we obtan Identty 15 (Vajda (37a)). k=2 f k 2 2 k = 1 f n+1 2 n.

7 PHASED TILINGS AND GENERALIZED FIBONACCI IDENTITIES 7 The same strategy can be appled to phased tlngs. Here, for convenence we assume the phase s determned by the frst tle (rather than the last). The phased dentty correspondng to equaton (2) s (3) n 1 G n+1 + G n + G k 2 n k 1 = 2 n (G 0 + G 1 ). The rght hand sde counts the number of ways to select a length n bnary sequence x and a phase p. From ths we construct a length n + 1 bnary sequence. If p s a domno phase, construct the sequence 0x; f p s a square phase, construct the sequence 1x. Interpret ths new n + 1-sequence as a Fbonacc tlng n the manner dscussed prevously, and assgn the tlng the phase p. By constructon, the phase s compatble wth the frst tle. (Recall that empty tlngs are assgned a domno phase.) A phased tlng of length n + 1 or n has a unque (x, p) representaton. For 0 k n 1, a phased k-tlng has 2 n k 1 representatons. Ths establshes equaton (3). Dvdng by 2 n gves Identty 16 (Vajda (37)). n 1 G k 2 k+1 = (G 0 + G 1 ) G n+1 + G n 2 n. 8. Dscusson The technques presented n ths paper are smple but powerful countng phased tlngs enables us to gve vsual nterpretatons to expressons nvolvng generalzed Fbonacc numbers. Ths approach facltates a clearer understandng of exstng denttes, and can be extended n a number of ways. For nstance, by allowng tles of length 3 or longer, we can gve combnatoral nterpretaton to hgher-order recurrences; however, the ntal condtons do not work out so neatly, snce the number of phases that the last tle admts do not correspond wth the ntal condtons of the recurrence. Another possblty s to allow every square and domno to possess a number of phases, dependng on ts locaton. Ths leads to recurrences of the form x n = a n x n 1 + b n x n 2. The specal case where b n = 1 for all n provdes a tlng nterpretaton of the numerators and denomnators of smple fnte contnued fractons and s treated n [2]. 9. Acknowledgment The authors gratefully acknowledge the referee for many valuable suggestons. References [1] A. T. Benjamn and J. J. Qunn, Recountng Fbonacc and Lucas Identtes, College Mathematcs Journal, to appear. [2] A. T. Benjamn, J. J. Qunn, and F. E. Su, Countng on Contnued Fractons, preprnt. [3] R. C. Brgham, R. M. Caron, P. Z. Chnn, and R. P. Grmald, A tlng scheme for the Fbonacc numbers, J. Recreatonal Math., Vol 28, No. 1 ( ) [4] L. Comtet, Advanced combnatorcs: the art of fnte and nfnte expansons, D. Redel Publshng Co., Dordrecht, Holland, [5] H. Prodnger and R. F. Tchy, Fbonacc numbers of graphs, Fbonacc Quarterly, Vol. 20 (1982) [6] S. Vajda, Fbonacc and Lucas Numbers, and the Golden Secton, John Wley and Sons, New York, AMS Subject Classfcaton Numbers: 05A19, 11B39.