GENERALIZATIONS OF ZECKENDORFS THEOREM. TilVIOTHY J. KELLER Student, Harvey Mudd College, Claremont, California

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1 GENERALIZATIONS OF ZECKENDORFS THEOREM TilVIOTHY J. KELLER Studet, Harvey Mudd College, Claremot, Califoria The Fiboacci umbers F are defied by the recurrece relatio Fi = F 2 = 1, F = F - + F 0 ( > 2) Every atural umber has a represetatio as a sum of distict Fiboacci umbers, but such represetatios are ot i geeral uique, J Whe costraits are added to make such represetatios uique, the result Zeekedorfs theorem [1], [5], Statemets of Zeckedorf f s theorem ad its coverse follow.calpha i s a iteger.,) Theorem. (Zeckedorf). Every atural umber N has a uique represetatio i the form N = Ew is where 0 < a < 1 ad if a = 1, the a = 0. Theorem. (Coverse of Zeckedorfs Theorem) ([1], [3]). Let { x f be a mootoe sequece of distict atural umbers such that every atural umber N has a uique represetatio i the form N = E*k x k i where 0 < <x < 1 ad if ca, ^ = 1» the a, = 0 9 The 95

2 96 GENERALIZATIONS OF ZECKENDORF f S THEOREM [Ja. {x L f = {F }. 1! L J 2 There are geeralizatios of Zeckedorfs theorem for every moor i toe sequece { x } of distict atural umbers for which x± = 1. The followig theorem is the first of may such geeralizatios. Theorem 1. Let the umbers x be defied by the recurrece relatio J x t = 1, x 2 = a, x = m-i x. + m? x 0 f > 2), 1 L s -1-2 ' ' where mj > 0, m 2 > 0, ad a > 1. The every atural umber N has a uique represetatio i the form N = 2>k*k where a >: 0 ad if a, ^ m l5 or,. = m 4 for 1 < i < p. i) ad p is odd 9 the a, < m 2 ; ii) p is eve, ad k ^ 1, the a, < m A ; iii) p is eve, ad k = 1, the a t < a. The special case m t = m 2 = 1, a = 2 is Zeckedorfs theorem, ad the case m 2 = 1, a = m A is a geeralizatio proved by Hoggatt.(5 e e p»89) Proof. We prove the existece of a represetatio by iductio o N. For N < x 2, we have N = Nx l9 Take N ^ x 2 ad assume represetability for 1, 2,, N - 1. Sice jx } is a mootoe sequece of distict atural umbers, ay atural umber lies betwee some pair of successive elemets of {x }. More explicitly, there is a uique ^ 2 such that x < N ^ x -. First let N < mix. There are uique itegers m ad r such that N = mx + r,

3 1972] GENERALIZATIONS OF ZECKENDORF f S THEOREM 97 where 0 < m < m t ad 0 < r < x. If r = 0, the N = mx, whereas if r > 0, the the iductio hypothesis shows that r is repre se table. Thus N is represetable* Now let N ^ m^x. Sice x,- = m-x + m 0 x - +1 I 2-1 for ^ 2 3 there are uique itegers m ad r such that N = m.x + m x - + r, 1-1 where 0 ^ m < m 2 ad O ^ r ^ x -. If r = 0, the L -1 N = m-x + mx., 1-1 whereas if r > 0 S the r is represetable. Thus N is represetable. Now use the iductio priciple. To prove the uiqueess of this represetatio, it is sufficiet to prove that x is greater tha the maximum admissible sum of umbers less tha & x accordig to costraits (i)-(iii). We prove this by iductio o. For = 1, this is obviously true. Take ^ 1 ad assume that the sufficiet coditio is true for 1, 2 9 -, - 1. From -1 E K X 2i-2 + ( m 2 " 1 ) x 2i-3> = E X 2i-l " E* 2 i-1 = X 2-1 " *' -1 E { m l X + { 2 i - l m 2 " 1 ) x 2i-2> = E x 2i " E x 2 i = x 2 - a, we obtai the idetities

4 98 GENERALIZATIONS OF ZECKENDORF'S THEOREM [Ja. x 2-l = L H x 2i-2 + ( m 2-1)x 2i-3> + X ' (1) 2 x 2 = 2 H X 2i-l + (m 2 " 1 ) x 2i-2 } + (a " 1 ) x l + 1 ' The iductio hypothesis together with (1) shows that x is g r e a t e r tha the maximum admissible sum of umbers l e s s tha x. Now use the iductio priciple. Theorem 1 ca be exteded to the case where the u m b e r s x a r e de fied by the r e c u r r e c e relatio Xi = 1, x = a (2 <= < q), q X = E m k V k fe - q), where m* 1 "> 0 5 m, ^ 0 for 1 < k < M? q, m -*- 0, ad 1 < a < a, for k q +1 1 < ^ q. Every atural umber N has a uique represetatio i the form N = S a i- Xi k k 9 1 where a. ^ 0 ad other costraits similar to those i T h e o r e m 1 a r e added. k F o r example, F if or -k+1.,- = m. k for 1 ^ k < p < q, the a _ < m. If F MJ -p+1 p p F = MJ q. the a,. < m. These costraits must be modified to fit the -q+1 q iitial coditios a. The proof of this extesio follows that of Theorem 1 ad uses the idetity

5 1972] GENERALIZATIONS OF ZECKENDORF'S THEOREM 99 q - r - 1 q-1 = 1L* m i y ^ x j + (m -1) y^ x. * - ^ k JLJI qi-r-k q L* *d q i - a - 1 q-r q-r-1 J q-r-1 fa a q-r -11 q-r L a q-r-l. 2 a q-ra, -1 q-r-1 q-r-2 \^ q~ r [ a q-r-lj ^ ^ / q - r - 1 J x t + 1 (0 < r < q) Statemets of two special c a s e s ad the proof of the secod oe follow. Theorem. (Dayki [3]). Let the u m b e r s x be defied by the r e c u r - rece relatio x = ( l < < q) 5 X = X., + X - 1 -q ( > q). The every atural umber N has a uique represetatio i the form N = E a k x k> where 0 ^ a, ^ 1 ad if a, i = 1> the a.,. = 0 for 0 ^ i < q - 1. M k k+q- 1 k+i T h e o r e m 2. Let the umbers x be defied by the r e c u r r e c e relatio x = (m + l / ' V < < q), x = m I P x, ( > q) M A»J - k The every atural umber N h a s a uique represetatio i the form N = J>, k x k ' i

6 100 GENERALIZATIONS OF ZECKENDORF'S THEOREM [ j a. where 0 ^ a. ^ m ad if a.,. = m for 1 < i < q, the a. < m. k k+i M ' k Proof. Followig the proof of Theorem 1, we prove the existece of a represetatio by iductio o N. F o r N ^ x, we have q-1 1 where 0 < a. < m. Take N ^ x ad a s s u m e represetability for 1, 2,, N - 1. There is a uique ^ q such that x ^ N < x -. Sice q-1 x ~~ m / x +1 La -k 0 for ^ q, there a r e uique itegers p, m T, ad r such that p - 1 N = m j> x. + m T x + r JLa -k - p, 0 where 0 < p < q, 0 < m T ^ m, ad 0 < r < x. If r = 0, the r 4J > _ p p - 1 N = m / ^ x 0, + m T x k -p """ I ~r ill.a. w h e r e a s if r ^ 0, the r is represetable. Thus N is represetable. Now use the iductio priciple. To prove the uiqueess of this represetatio, we prove that x is g r e a t e r tha the maximum admissible sum of umbers l e s s tha x a c c o r d - ig to the costraits by iductio o. F o r 1 < ^ q, we have m # X ) x k = m E (m + 1)k " 1 = (m + 1)11 " 1 " X ^ (m + 1)_1 = x ' 1 1

7 1972] GENERALIZATIONS OF ZECKENDORF'S THEOREM 101 Take /* q ad assume that the sufficiet coditio is true for - q. The x = m x. + (m - l)x + x E, -k -q - The iductio hypothesis shows that x is greater tha the maximum admissible sum of umbers less tha x. Now use the iductio priciple. umbers x Zeckedorf s theorem ca be further geeralized to cases where the are defied by recurrece relatios with egative coefficiets. Theorem 3. Let the umbers x be defied by the recurrece relatio J x t = 1, x 2 = a 5 x = m-x - - m 0 x 0 ( > 2), N where 0 < m 2 < m^ ad a > m 2. uique represetatio i the form The every atural umber N has a N = E*k x k where 0 < a, < ^ for k > 1, 0 < a t < a 9 ad if ^ k + D + 1 = m i -!> \ + i = mi - m 2-1 for 1 < i < p 3 ad (i) k > 1, the a. < m-j_ - m 2 ; (ii) k = 1, the c^ < a - m 2. the idetity The proof, which will ot be give, follows that of Theorem 1 ad uses

8 102 GENERALIZATIONS OF ZECKENDORF'S THEOREM Ja x = (m t - l)x _ 1 + (m t - m 2-1) ^ x. + (a - m 2 - l)x t The coverse of Zeckedorf's theorem ca be geeralized to iclude as special cases the coverses of the geeralizatios of Zeckedorf's theorem give so far. r i Theorem 4. Let {x } be a mootoe sequece of distict atural umbers such that every atural umber N has a uique represetatio i the form N = I X x k where a, ^ 0 ad other costraits o {<\} are added such that the reprepresetatio of x is itself. The {xj- is the oly such sequece. r V r i 00 Proof. Assume the sequeces {x } ad {y } both satisfy the hypotheses, where {x r = {y } in L J A lj J i ad y - ^ XN+1 * T e 3^+1 ^ a s a u iq u e represetatio as a sum of umbers x, each of which i tur has a uique represetatio as a sum of umbers y, where ^ N. O the other had, y N - obviously represets itself ad, thus, y N + 1 has two represetatios i terms of umbers y. This cotradicts the uiqueess of represetatio, ad we coclude that w : - <* Theorem 4 does ot iclude the coverse of the followig geeralizatio of Z ec kedo rf's theorem. Theorem (Brow [2]). Every atural umber N has a uique r e p r e- setatio i the form of [Cotiued o page 111. ]