are fractions which may or may not be reduced to lowest terms, the mediant of ( a

Transcription

1 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS WITH MEDIANTS HAROLD REITER AND ARTHUR HOLSHOUSER Abstrct. The Stern Brocot tree is method of generting or orgnizing ll frctions in the intervl (0, 1 b strting with the endpoints 0 1 nd 1 nd repetedl ppling the medint opertion: 1 m ( b, c d +c. A recent pper of Ailm considers two generliztions: One is to ppl the medint opertion strting with n rbitrr intervl ( b, c d (the frctions must be non-negtive, nd the other is to llow rbitrr reduction of generted frctions to lower terms. In the present pper we give simpler proofs of some of Ailm s results, nd we give simpler method of generting just the portion of the tree tht leds to given frction. 1. Introduction The Stern Brocot tree is method of generting ll non-negtive frctions reduced to lowest terms nd without repetition (see for emple [?, pp ]. The clssicl Stern Brocot tree strts with the two frctions 0 1 nd 1 1 nd repetedl pplies the medint opertion: Definition 1.1. If 0 b < c d re frctions which m or m not be reduced to lowest terms, the medint of ( d c is denoted b m b, d c nd is defined b m ( b, c d +c. The frctions cn be orgnized in binr tree, where ech frction hs two offspring, both formed s medint of tht frction nd the nerest frction to its left or right. The tree is in two smmetric hlves becuse the first medint is 1 1 nd s frction is generted on the left, its reciprocl is generted on the right. Therefore we could lso think of it s w of generting ll the frctions in the intervl (0, 1 strting with the frctions 0 1 nd 1 1. The Fre series F n is formed b pruning the tree to omit ll frctions with denomintors greter thn n; see [?, pp ] Historicll the Stern Brocot tree rose s method of pproimting frctions with lrge denomintors closel b frctions with much smller 1

2 H. REITER AND A. HOLSHOUSER denomintors, in prticulr in the contet of designing collections of gers for clocks; see for emple [?]. A recent pper b Ailm [?] considers the generliztion of this process strting with rbitrr non-negtive frctions 0 b < c d. One intriguing feture of the clssicl Stern Brocot tree is tht the medints re lws in lowest terms nd do not need to be reduced. In the generliztion this is no longer true, even if the strting frctions re in lowest terms; see Emple?? below. Different trees re generted ccording to whether we reduce some or ll of the frctions to lower or lowest terms. We distinguish two cses or methods: The es cse never reduces n medints, nd the hrder cse which m reduce some medints. In this pper it does not mtter whether the end points re reduced to lowest terms or not. However, for convenience onl we will reduce b, c d to lowest terms. In the es first prt of this pper (Sections???? we gree not to reduce n of the medints to lower terms if the medint is not lred reduced to lowest terms. Then in the hrder prt of the pper (Section??, we generlize the theor so tht the reder cn decide for himself which medints he wishes to reduce to lowest terms nd lso decide if he wishes to prtill or completel reduce n of the medints to lowest terms. The second prt is hrder to prove. Definition 1.2. We cll ( b, d c bc d the delt-vlue or the -vlue of the intervl ( b, d c. ( We note tht if +c is not reduced to lowest terms then b, +c ( +c, c d ( d c bc d. We see tht if b, d c bc d 1, then +c is reduced to lowest terms since b ( + c (b + d bc d 1. Also, it is es to show tht 0 b < +c < c d. Tht is m ( b, d c ( b, c d. In this pper we show tht the Stern Brocot tpe frctions tht re generted from ( b, c d uniquel compute ll rtionl numbers ( b, c d nd this is true whether the medints re reduced to lowest terms or not. When ( d c 0 1, 1 1, the medints generte the Stern Brocot frctions in (0, The pln. Suppose 0 b < c d re frctions reduced to lowest terms lthough this ssumption is not needed. We first show tht the Stern Brocot medints tht re generted b b, c d uniquel compute ll rtionl numbers q [ b, d] c in the es cse, where ll medints re not reduced to n lower terms. We do this b reducing the problem to the known result when b 0 1, c d 1 1. This is the result for the clssicl Stern Brocot tree nd is well-known; see for emple [?, Theorem 1] or [?, pp ]. 2 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2

3 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS We lso put n upper bound T on the number of steps it tkes for the telescoping serch pth to rech n given q [ b, d] c in this ese cse. Of course, T depends on q. Using this es cse nd the upper bound T for ech q [ b, c d] s our min mchiner, we esil etend the es cse to the hrder cse where the medints cn be reduced to lower terms in n rndom nd rbitrr w including prtill reducing to lower terms. The ide tht we use is tht when we go down telescoping serch pth to tr to find given q [ b, c d] the medints in the serch pth cn onl be reduced to lower terms t most T times where T is known upper bound. The serch pth cn be s long s it tkes to find q. When the numbers of steps between these medint reductions is s big s T we cn use the es cse with its upper bound T to find the given q [ b, d] c. We will lso give crude fst lgorithm t the end of Section?? which will llow the reder to completel skip the detiled lgorithm tht follows Section?? if he wishes. Insted of using the medints of ( b, d c, we hve develop n nlogous where e f is the b, d c tht hs the smllest possible size e + f. When theor in which we use e f ( d c in the plce of m b, c d unique frction in ( bc d 1, we cn show tht e f equls ( + c + (b + d. ( b, c d course, the size of the e f m ( b, c d +c nd, of This sttement hs ver es proof nd it gives trivil solution to the problem under discussion when bc d 1. It lso gives ver es proof to Theorems 1, 2 of [?]. This fct illustrtes tht the serch lgorithm given in [?] is ver different from the one in this pper. In the lst section we use our mchiner to clssif 0 < b m n < < c d where bc d 1 nd n m Modified Stern Brocot Frctions We first define the levels of the modified Stern Brocot frctions, then prove some of their properties Construction. If 0 b < c d re frctions reduced to lowest terms, we define the vrious levels of the modified Stern Brocot frctions s follows. MISSOURI J. OF MATH. SCI., FALL

4 H. REITER AND A. HOLSHOUSER We illustrte the pttern first nd then we formlize this pttern. p 8 p 9 p 10 p 11 p 12 p 13 p 14 p 15 p 4 p 5 p 6 p 7 p 2 p 3 b p 1 c d Fig. 1 Levels of Modified Sterm-Brocot Frctions In Sections????, we gree not to reduce n of the medints to lowest terms. In other words, we re computing smbolicll. Figure?? shows picture of the following. We denote L 0 { b, d} c nd cll L0 level 0. L 1, level 1, consists of the single point p 1 m ( b, c d +c. Note tht 0 b < p 1 < c d. Using this ordering b < p 1 < c d we define level 2, denoted L 2, s follows. L 2 consists of the two points p 2 m ( b, p ( 1 m b, +c 2+c 2 ( b, p 1 nd p 3 m ( ( p 1, c d m +c, c d +2c b+2d ( p 1, c d. Note tht b < p 2 < p 1 < p 3 < c d. Using this ordering b < p 2 < p 1 < p 3 < c d we define level 3 s follows. Level 3, denoted L 3, consists of the four points p 4 m ( ( b, p 2 m b, 2+c 2 3+c 3 ( b, p 2 nd p5 m (p 2, p 1 3+2c 3b+2d (p 2, p 1 nd p 6 m (p 1, p 3 2+3c 2b+3d (p 1, p 3 nd p 7 m ( p 3, c d +3c b+3d ( p 3, d c. Note tht p 4 < p 5 < p 6 < p 7 nd b < p 4 < p 2 < p 5 < p 1 < p 6 < p 3 < p 7 < c d. Using this ordering we define in similr w the eight points p 8 < p 9 < p 10 < p 11 < p 12 < p 13 < p 14 < p 15 of level 4. We continue this construction pttern to crete levels L 1, L 2, L 3, L 4, L 5,, where ech level i 1 hs 2 i 1 elements. We emphsize tht in 4 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2

5 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS this section we re not reducing our frctions p 1, p 2, p 3, to lowest terms. We now formlize this pttern. First from the construction pttern, we note tht the level i 1 points lternte with the points of i 1 l0 L l. Indeed, the members of level L i re the medints of the pir of consecutive points of the ordered set i 1 l0 L l. Also, b this definition of the construction pttern nd b ( simple induction, we cn see tht ech L i, i 2, is written s m, + + where one of, is level i 1 point nd the other, lies in the set i 2 k0 L k. This follows since the level i 1 points, i 2, lternte with the points of i 2 k0 L k. Fig. 1 should mkes this cler. As n emple of the lterntion we note tht the level 4 points re p 8 < p 9 < p 10 < p 11 < p 12 < p 13 < p 14 < p 15 nd these eight points lternte with the members of { k3 k0l k b < p 4 < p 2 < p 5 < p 1 < p 6 < p 3 < p 7 < d} c s follows: b < p 8 < p 4 < p 9 < p 2 < p 10 < p 5 < p 11 < p 1 < p 12 < p 6 < p 13 < p 3 < p 14 < p 7 < p 15 < c d. We now denote level i 0 b L i nd we denote i k0 L k L i. Note tht for i 1, L i L i ( i 1 k0 L k Li L i 1. Also, we denote k0 L ( k L. It is es to see tht the members of L re ll distinct since m, (, nd from this we see tht the members of L re ll distinct since the members of ech L i L i 1 φ Properties. Theorem 2.1. Suppose L i i k0l k { b p 0 < p 1 < p 2 < p k c }, i 0 d, Then ( p t, p t+1 bc d for ll pirs pt, p t+1 of consecutive members of the ordered set L i. Proof. We cn ssume b induction tht if { L i 1 b p 0 < p 1 < p 2 < < p s d} c then ( p t, p t+1 bc d. First, note from the bove tht Li L i L. Now the set L i consists of the set of medints m ( p t, p t+1 where p t, p t+1 MISSOURI J. OF MATH. SCI., FALL

6 H. REITER AND A. HOLSHOUSER rnge over the consecutive members of L i 1. Now if ( p t, p t+1 bc d, then we know tht ( p t, m ( ( ( p t, p t+1 m p t, p t+1, p t+1 ( ( ( p t, p t+1 bc d. This is becuse,, + + ( + +,. Now { L i L i L i 1 b p 0 < p 1 < p 2 < < p k d} c. Therefore, ( p t, p t+1 bc d. Therefore, if pt, p t+1 ( then, bc d nd this implies gcd (, nd gcd (, bc d. We stte this gin in Section??. Of course, gcd mens gretest common divisor. In Sections???? we re computing smbolicll nd not reducing the medints to lowest terms so it is obvious tht for i 1 nd i rbitrr tht ech p t L i \ { b, } c d stisfies pt c φb+θd where φ, θ {1, 2, 3, }. Also, b 1 +0 c 1 b+0 d nd c d 0 +1 c 0 b+1 d. Theorem 2.2. We hve p t c φb+θd prime. where (φ, θ must lso be reltivel Proof. To see this, let p t c φb+θd nd p t+1 c where p φb+θd t < p t+1 re consecutive members of the ordered set L i nd where φ, θ, φ, θ {1, 2, 3, }. We know tht ( p t, p t+1 bc d. B clcultion, ( ( p t, p t+1 φθ θφ (bc d bc d since ( p t, p t+1 bc d. Therefore, φθ θφ 1 which implies tht both pirs (φ, θ nd ( φ, θ re reltivel prime. B the sme clcultions, it is es to show tht if φ, θ, φ, θ {1, 2, 3, } nd both of (φ, θ, ( φ, θ re reltivel prime then c φb+θd c is true if φb+θd nd onl if (φ, θ ( φ, θ. This is becuse φθ θφ 0 if nd onl if θ φ θ φ which is true if nd onl if (φ, θ ( φ, θ. Since the members of L k0 L k re distinct, we see tht if c φb+θd c re two different members of L φb+θd, then (φ, θ ( φ, θ. In other words, if c φb+θd L then the ordered pir (φ, θ, φ, θ {1, 2, 3, }, (φ, θ re reltivel prime, cn pper t most one time, in L. Since 0 < bc d, we see tht c φb+θd c is true if nd onl if 0 < φθ θφ which is true φb+θd if nd onl if θ φ < θ. φ 6 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2

7 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS Theorem 2.3. We hve { L \ b d}, c { } φ + θc : φ, θ {1, 2, 3, }, φ, θ re reltivel prime. φb + θd In other words, L \ { b, d} c picks up ll ordered pirs (φ, θ where φ, θ re reltivel prime nd φ, θ {1, 2, 3, }. To see this on [0, 1], let ( d c 0 1, ( 1 1. Note tht 0 1, , 1 1. From the theor of Stern Brocot frctions, we know tht when ( b, d c ( 0 1, 1 1 then L \ { 0 1, 1} 1 {q : 0 < q < 1, q is rtionl nd q is reduced to lowest terms}. The comment t the end of Section?? shows how we proved this. Also, see Theorem 1, [?]. Since ( 0 1, ll medints of L \ { 0 1, 1} 1 re utomticll reduced to lowest terms. Note tht L \ { 0 1, 1} 1 picks up ll rtionl numbers q ( 0 1, 1 1, where is reduced to lowest terms. Now when ( d c 0 1, 1 1, we see tht c φb+θd φ 0+θ 1 φ 1+θ 1 θ where φ, θ {1, 2, 3, } nd (φ, θ re reltivel prime nd θ is utomticll reduced to lowest terms. Now ech rtionl number (0, 1, 0 < < where is reduced to lowest terms, cn be uniquel represented b θ where θ, φ nd θ, φ {1, 2, 3, } nd (θ, φ re reltivel prime. Conversel, if θ, φ {1, 2, 3, } re rbitrr nd (θ, φ re reltivel prime then θ θ (0, 1 nd is reduced to lowest terms. Since L \ { 0 1, } 1 1 picks up ll rtionl numbers, 0 < <, where is reduced to lowest terms, it is es to see tht { 0 L \ 1, 1 } 1 In other words, L \ { 0 1, 1 1 nd (θ, φ re reltivel prime. { θ : θ, φ {1, 2, 3, }, θ, φ re reltivel prime φ + θ } picks up ll θ }. where θ, φ {1, 2, 3, } Proof of Theorem??. Using ( d c in the plce of d c 0 1, 1 1 it is es θ c to see tht becomes φb+θd so tht L \ { 0 1, 1} 1 becomes { L \ b d}, c { } φ + θc : θ, φ {1, 2, 3, }, θ, φ re reltivel prime. φb + θd } nd L \ { 0 1, } 1 1 In other words, the smbolic clcultion of L \ { b, c d re ectl the sme. We now give second proof of this. We show tht ll frctions θ+φc ( θb+φd L b, d c when gcd(θ, φ 1. The proof is b mthemticl induction on θ +φ. It is obvious for n 1. Therefore, suppose it is true for θ +φ n 1. MISSOURI J. OF MATH. SCI., FALL

8 H. REITER AND A. HOLSHOUSER We show tht it is true for θ + φ n. Consider the two subtrees ( +c nd, c d. B smmetr, we m suppose tht θ φ. Then θ + φc (θ φ + φ( + c θb + φd (θ φb + φ(b + d ( b, +c where gcd(θ φ, φ 1. Now (θ φ + φ θ n 1. Therefore b the induction hpothesis ( θ φ + φ( + c (θ φb + φ(b + d L b, + c. b + d Therefore, θ + φc ( θb + φd L b, c. d Min Theorem. Suppose 0 b < c d where b, c d re reduced to lowest terms nd ( d c is n rbitrr rtionl number in b, d c reduced to lowest terms. Then L \ { b, d} c. Note 2.4. Recll tht the members of L \ { b, d} c re not being reduced to lowest terms. However, when we s tht L \ { b, } c d where is reduced to lowest terms, we men tht m n where m n L \ { b, d} c nd m n fter m n is reduced to lowest terms. Proof of Min Theorem. Let c φb+θd {1, 2, 3, } nd (θ, φ re reltivel prime. Now true if nd onl if φ (b θ (c d. Therefore, φ θ c d c φb+θd where we wish to compute θ, φ is true if nd onl if φb + θd φ + θc. This is b where c d > 0, b > 0 since b < < c d. We now gree to reduce φ θ to lowest terms. Our proof of the Min Theorem is lmost ectl the sme s the proof of Theorem 2 of [?], lthough the ltter theorem sttes 1 s hpothesis. This mens tht the proof of Theorem 2, [?] cn esil prove fr more thn wht Theorem 2, [?] ctull sttes. Also, once Theorem 2, [?] hs esil been enhnced, then Theorem 7 [?] (which is our Min Theorem cn esil be proved in crude w b using our Sections??,??. See the crude fst lgorithm t the end of Section?? to see how we do this. 8 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2

9 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS 3. Some Useful Lemms Lemm 3.1. Suppose 0 b < c d re frctions reduced to lowest terms. Let L n where L n is the level n of the modified Stern Brocot frctions nd n 1. As lws, is not reduced to lowest terms. Then n. { } +c Proof. We prove this b induction on n. Now L 1 nd b + d 2 > 1. As stted previousl in Section??, ech L n, n 2, cn be written s ( m, + + where one of, is member. L n 1 nd the other, is member of L n 2. B smmetr ssume tht L n 1, L n 2. B induction n 1. Also, 1. Therefore, (n n. Lemm 3.2. Using the hpothesis of Lemm??, Suppose L n, n 1, where s lws is not reduced to lowest terms. Then gcd (, nd gcd (, where bc d. Since n 1 is rbitrr this implies tht ech L \ { b, d} c stisfies gcd (, nd gcd (,. Proof. The proof ws given in Section?? nd used the fct tht if n 1 nd L n { b p ( 0 < p 1 < p 2 < < p k d} c, then pi, p i+1 bc d. Also, L n L n. Let p i, p i+1. Then ( ( p i, p i+1, bc d. Therefore, gcd (, nd gcd (,. Lemm 3.3. Suppose ( b, c d where 0 b < c d re reduced to lowest terms nd where is frction tht is now reduced to lowest terms. Then L 1 L 2 L 3 L where bc d. We s tht L 1 L 2 L 3 L b the mening of Note?? since is reduced to lowest terms nd the members of L 1 L 2 L re not reduced to lowest terms. Proof. B the Min Theorem, we know tht i1 L i L \ { b, d} c (b the mening of Note??. Suppose n + 1. We show tht / L n. Let / L n, where n + 1, nd is rbitrr nd where s lws the members of L n hve not been reduce to lowest terms. We show tht b Note??. Now b Lemm??, n +1. Also, b Lemm??,( gcd (, (. Therefore, when is reduced to lowest terms, we hve / nd gcd(, gcd(, gcd(, > (where is reduced to lowest terms. Tht is gcd(, >. Therefore, if L n nd n +1, then. This implies L n when n + 1. Therefore, L 1 L 2 L. MISSOURI J. OF MATH. SCI., FALL

10 H. REITER AND A. HOLSHOUSER Appliction 3.4. Suppose ( b, c d where 0 b < c d re reduced to lowest terms nd is frction tht is reduced to lowest terms. Also, suppose bc d 1. Then L 1 L 2 L since. Thus, ll frctions ( b, c d where is reduced to lowest terms nd n re picked up in L 1 L 2 L n. This is ver importnt in the theor of Stern Brocot frctions where ( d c 0 1, 1 1 nd An Algorithm for Computing ( b, d c s member of L \ { b, } c d We now define telescoping sequences. Let 0 b < c d where b, c d re reduced to lowest terms. The reder needs to drw picture of the following. Definition 4.1. A telescoping sequence is sequence 0, 0, 1, 2, 3, where 0 b, 0 c d, 1 m ( 0, 0 p 1 +c. Also, 2 m ( 0, 1 or 2 m ( 1, 0 where we mke binr choice for 2. In generl if i m (,, <, then i+1 m (, i or i+1 m ( i,. Note gin tht we mke binr choice for ech of 2, 3, 4,. Also, we know tht 0 b, 0 c d re on level L 0. Also, 1 p 1 +c is on level L 1. Also, 2 is on level L 2. Also, 3 is on level L 3. We now show tht ech i, i 1, is on level L i. Indeed, the following dditionl fcts re es to prove b definition nd b induction. We simpl observe tht the simple pttern tht we now give repets itself over nd over s we go higher. (The reder m need to review the definitions of L i, L i, i 1. The pttern is the following. If i+1 m (,, <, then i+1 L i+1 (tht is, i+1 is on level L i+1. Also,, L i nd, re consecutive members of the ordered set L i. Also, < i+1 < re consecutive members of the ordered set L i+1 where L i+1 L i+1 L i. Now, i+2 m (, i+1 or i+2 m ( i+1,. In either cse, i+2 L i+2. If i+2 m (, i+1, then < i+2 < i+1 re consecutive members of the ordered set L i+2 where L i+2 L i+2 L i+1 nd if i+2 m ( i+1,, then i+1 < i+2 < re consecutive members of the ordered set L i+2. This pttern repets itself s we go higher. In conclusion, we lso note tht ech i is utomticll ssocited with n intervl (, where i m (,, i (,. Also, if i+1 m (, i we ssocite i+1 with the intervl (, i nd if i+1 m ( i,, we ssocite i+1 with the intervl ( i,. Problem 4.2. Suppose 0 b < c d re given frctions reduced to lowest terms nd m n ( b, d c is given rtionl number reduced to lowest terms. 10 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2

11 GENERATING STERN BROCOT TYPE RATIONAL NUMBERS Show how to clculte step b step the construction of m n L \ { b, } c d. Of course, we s tht m n Note?? of Section??. s member of L \ { b, c d} b the mening of Solution. Of course, b the Min Theorem we know tht m n L \ { b, c d}. We now clculte telescoping sequence 0 b, 0 c d, 1 m ( 0, 0, 2, 3,, t 1, t such tht for ech i if i m (,, <, (which mens tht i is ssocited with (, then m n (,. If m n i then, of course, we re done. If m n i, then we define i+1 m (, i if m n (, i nd i+1 m ( i, if m n ( i,. Likewise, if m n i+1 then we re done. If m n i+1 we then clculte i+2 the sme w tht we clculted i+1. This process will clculte telescoping sequence 0, 0, 1, 2,, t 1, t where ech i is on level L i nd, b Lemm?? of Section??, we know tht m n t for some t where 1 t n, bc d. This is becuse m n L 1 L 2 L n, bc d. Note 4.3. Of course, usull we will rech m n t much sooner thn t n since the denomintors of most members L n re much bigger thn n. Emple 4.4. Clculte ( 1 3, 3 5 s member of L \ { 1 3, 5} Note tht we do not reduce Solution , 0 3 5, 1 m ( 1 3, to lowest terms ( 1 3, 3 5, 1 m ( 1 3, ( 1 3, 4 8, 2 m ( 1 3, ( 5 11, 4 8, 3 m ( 5 11, ( 9 19, 4 8, 4 m ( 9 19, ( 13 27, 4 8, 5 m ( 13 27, ( 13 27, 17 35, 6 m ( 13 27, Note tht in the lst step , 5 11, 9 19, 13 27, 17 35, we must reduce A Crude Fst Algorithm to lowest terms. Suppose tht when we crr out the bove telescoping serch lgorithm tht we gree to reduce the serch medints to lower terms in n possible w tht we choose including lws reducing or never reducing or prtill reducing. We cn use induction on bc d to show tht the serch MISSOURI J. OF MATH. SCI., FALL

12 H. REITER AND A. HOLSHOUSER lgorithm will still pick up the rtionl number q in [ b, d] c for which we re serching. If 1 then the result is obvious since ll medints re lred reduced to lowest terms. If we never reduce our serch medints to lower terms then we know tht we will find. If we reduce serch medint to lower terms before we find the induction on will show tht we must eventull find. Note tht if (q, q nd q is reduced to q then (q, q < (q, q. If we find before we reduce n serch medints to lower terms there is nothing to prove since we hve found. Acknowledgements The uthors wish to thnk the referee for his etremel helpful informtion which ws unknown to us. We lso thnk Professor Ben Klein of Dvidson College for his help. References [1] D. Ailm, A generlized Stern-Brocot Tree. Integers 17 (2017, #A19, mth.colgte.edu/~integers/vo117.html [2] D. Austin, Trees, Teeth, nd Time: the mthemtics of clock mking. Americn Mth Societ Feture Column, December 2008, feture-column/fcrc-stern-brocot [3] R. L. Grhm, D. E. Knuth, O. Ptshnik, Concrete Mthemtics, Second Edition. Addison-Wesle, Upper Sddle River, New Jerse, MSC2010: 11B57 Ke words nd phrses: Stern Brocot tree, Fre sequence Deprtment of Mthemticl Sciences, Universit of North Crolin Chrlotte, Chrlotte, NC 28223, USA E-mil ddress: hbreiter@uncc.edu 3600 Bullrd St, Chrlotte, NC 28208, USA 12 MISSOURI J. OF MATH. SCI., VOL. 28, NO. 2