Theorem 1: R(F n+l ~2-M)=R(F n^m) y 0<M<F n u n> ] 47

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1 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS, COUNTED BY RECURSIVE FORMULAS Marjorie Bicknell-Johnson 665 Fairlane Avenue, Santa Clara, CA 9505 Daniel C. Fielder School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA (Submitted April 997-Final Revision July 997) 0. INTRODUCTION Let R{N) be the number of representations of the nonnegative integer A^asa sum of distinct Fibonacci numbers. For N = F n -l 9 n>\, the Zeckendorf representation, in which no two consecutive Fibonacci numbers appear in the sum, is the only possible representation, and R{F n -) =, as proved by Carlitz [] and Klarner []. The sequences {b n -}, b n+l = b n +b n _ l9 arise as a generalization, having the property that R(b n -l) = R(b n+l -l) = k for all sufficiently large n (see [] and []). The generation of the specialized and related sequence,, 8, 6,,..., i» whose 7 th term is the least N such that n = R(N), spurred efforts to find recursive relationships for the values R(N) and ways to compute R(N) for large values of N. Some authors have used T(N) and some R(N) in counting representations; we will use R(N) for the number of ways to represent N as a sum of distinct Fibonacci numbers (without i*j) and T(N) for the number of representations if both F x and F are used. In our notation, Carlitz and Klarner both givei?(f ) = [«/], «>, where [x] is the greatest integer inx. Since T(N)=R(N)+R(N-l), we have concentrated on formulas for R(N). Earlier authors have used generating functions and combinatorics to develop and prove representation theorems. In this paper we concentrate on properties of the integers whose representations are being counted. We prove Conjectures,, and from [] as well as writing formulas for R(MF k ) and R(ML k ), M>, and solving R(N)=mR(N-l)-q for integers M 9 m, mdq.. THE SYMMETRIC PROPERTY AND A BASIC RECURSION The most obvious property in a table of R(N) is the palindromic subsequences it contains, beginning and ending with, for N in the interval F - < N < F n+l - ; i.e., when 0 < M <F n _ x, n>, R(F n+l -l-aj)=r(f n -l + M). () Since these values R(N) are symmetric about the center of each palindromic segment, we only have to compute the values of the first half of the interval. Symmetric property () is a variation of Theorem, whose results appear in Klarner [5], as specialized for the Fibonacci sequence Theorem : R(F n+l ~-M)=R(F n^m) y 0<M<F n u n>. 999] 7

2 THE NUMBER OF REPRESENTATIONS OF JV USING DISTINCT FIBONACCI NUMBERS Values for R(N) for 0 < N < 60 N 0 i R(M JV i 5! 6 7 ' - 8, 9 0 *w 5, 5 N!., ! ' : 5. R(N) i. ' J N R(N) It is a simple matter to compute a table for R(N) from generating functions for small N, but as JV gets larger, the computer's memory will eventually be exceeded. We have calculated R(N) for < N < 57,5 and have capabilities of calculating individual values for R(N) for very large TV; for example, R (,000,000,000) = 665. We have listed {A n } for </?<0. But to "study the mysteries of {A n } or to compute R{N) for large Nby hand, we need some recursive relationships. Klarner [5] proved Theorem for generalized Fibonacci numbers. Theorem (Basic Recursion Formula): If F n < M < F n+l -, then R(M)=R(F n+l --M)+R(M-F n X n>. () Lemma : If F n < M < F n+l -, then R{M-F n ) is the number of representations of Musing F m while the number of representations of M using F n _ x is R(F n+l - - M). Proof: The largest Fibonacci number in Mis F n. R(M) is the sum of the number of representations of M that use F n and the number of those that use F n _ x. Since M < F n+l -, no representations of Muse both F and F n _ Y \ else M>F n+l, There are no representations of M that use neither F n nor F n _ u since F - = F n _ + F n _ F + F <M. Note that M = F n +M x, where the largest possible Fibonacci number in M x is F n _ ; else M could contain i^+. The number of representations of M that use F is RiM^ =R(M-F ) since F n is added to each possible representation of M x to make a representation of Musing F w. To list representations of Musing F _i, if we write M = F _ x +F n _ +Mj and then list representations of M u there can be a repetition of terms, such as F n _ appearing twice, so we need sums using disjoint sets of Fibonacci numbers. Representations of (F n+l --M) = (F n _ { + F n _ + + F + F )-M will use a set of Fibonacci numbers disjoint from those selected to represent M. Thus, R(F n+l - - M) must give the number of representations of M that use F _ x by examining Theorem. In counting by hand, R(M)=R(M-F )+R(M-F _ l ) if M-F nr. l <F n _ v For example, = + = +0, and i?() = R() +i?(0). If M-F n^ > F _? an adjustment must be made; 8 [FEB.

3 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS 0 = + 9 = + 7 = + ( + ), and R(0) = R(?)+R(7) -R(). Lemma makes this counting correction. We take R(0)='l and R(K) = 0 when K<0 in Lemmas through 6, and [x] denotes the greatest integer in x. Lemma : If F <M<F + -, then R(M)=R(M-F ) + R(M-F n _ x )-R(M-F _ X ); R(F n+l --Ad) = R(M-F n _ x )-R(M~F n _ x ). Proof: R(M) is the number of representations of Musing F n plus the number of representations of Musing F n _ x corrected for the number of representations of (M-F n _ x ) using F _ u is any exist. A second way to write the representations of M that use F n _ x is to write M = F n _ x + (M-F n _i) and observe that the number of representations that use F n _ x is R(M - F n _ x ) if F _ x is not used in representing (M - F n _ x ). If M > F n _ x, R(M - F _ l ) is the number of representations of ( M - F ^ ) using F n _ h since M-F n _ x = F n _ x + {{M-F n _ x )-F n _ x ). Thus, the representations ofm using F n _i are counted by [R(M - F n _ x ) R(M F n _ x )\, which count appeared in Lemma asi?(f w + --M). D Lemma : R(F + K) = R(F _ l --K)+R(K),0<K<F _ l -. () Lemma is another form of Theorem, while Lemma results when M = K + F n in (), and is useful in computation. For example, let K =, R(K) = 5; since 0< K<F n _ x -, take»> 0.» = H = «= «= 6 fl( + )=.R(87-)+/?() = 8 + 5; 7?(68) =, tf( + ) =./?(-)+i?() = 0 + 5; R(57) = 5, R( + 77).= R( - ) +R() =. + 5; /?(0) = 8, rt( + 987) = /?(608-)+/?() = 8 + 5; #(0) =, where we recognize, 68, 57, 0, and 0 as members of our specialized sequence {A }. J eiftifta * R(M) = R(M- F n )+R(M - F B _,), F < M < F + F _ -. Proof: Because F n _ x = F + F _, R(M- F _ l ) = 0 in Lemma throughout the interval chosen. Lemma 5: R(N) for the interval F <N< F n+x - is given by: R(F + K) = R(F n _ + K) +R{K), 0<K<F _ -; R(F + K) = R(K), F _ <K<F n _ -\; (5) i?(f + ^ ) = /?(F + --X), F n _ <K<F _,-\. Proof: Let M = F n + K in Lemma and use Theorem to write the first and last F n _ values of R(N). Let F _ +p = K\n Lemma, followed by application of Theorem since 0 < p < F _ : R(F n + /v_ + p) = R(F^ - - (F _ + p)) +R(F _ + p) = R(F n _ --p) + R(F n _ + p) = R(F _ + p)+r(f r,_ +p). Thus, R(F n + K) = R(K) when F _ <K<F n _ -\. D l } 999] 9

4 THE NUMBER OF REPRESENTATIONS OF FUSING DISTINCT FIBONACCI NUMBERS Lemma 6: R(F + K) = R(F n _ +K)+R(K)-R(K-F _ ), 0<K<F _ v (6) Proof: For 0 < K < F _ t -, take M = F + K in Lemma, so that M - F _ l = (M- F )+ (F n -F n _ ) = K-F n _. Then let K = F n _ x - in the expression above, using R(F -) =. Finally, take K = F _ x, using R(F n+ ) = [(n + )/] = R(F ) + from [] and [].. SPECIAL VALUES FOR R(b - ) AND R(b n ) Recursive sequences {b n -}, b n+l = b n +b n _, have R(b n - l)=r(b n+l -) = k for n sufficiently large (see [] and []). We can write sequences for which R(N-l) = k, a given constant, as indicated in the following example. Say k = 5 is given. Find a particular value, i.e., i?() = 5. Write + = 5 = + + in Zeckendorf form, or R() = R(F S + F + F l -) = R(F 8 +F +F -) = 5. These are the first terms, when F n =, in sequences we seek. Thus, R(F +7 +F n+ +F -l) = 5 = R(F n+7 +F n+ +F n+l -lx The symmetric property gives R(F n+ -l + M)=R(F n+% -\-M) we can write n>\. R(F + s-l-(f + +F ))=R(F +7 +F n+s + F +l -l) = 5,» >. Since R(F l0 ) = R(F l0 + -) = 5, again using the symmetric property, R(F +9 +F n -\)= R(F +9 + F n+l -) = 5, n >, *( + io ~F -l) = R(F +l0 - F +l -) = 5, n >. Since R(F k ) = R(F k+l ) = k,we can derive in a similar way, for n > : R(F k - l+ +F n -d = k =R(F k _ l+ +F +l -); R(F k+ -F -\) = k= R(F k+n -F n+ -), forn>. = 5 for M = F n+ +F n, so that For a given value of k, there are many infinite sequences such that R(b -) = k. All ways of writing infinite sequences such that R(b -) = k, for k =,,, were given by Klarner [] as Some useful equivalent statements are R(F -l) =R(F +l -l) =; *(^+ + F -) = R(F n+ + F +l -) = ; R(F +5 + F -l)= R(F +5 + F +l -) - ; *(^+6 - ^ -) = *(^ F + -) -. R(F n+ -l) = R(L n+ -l) =; i?(f + -l) =/?(F + -l) =; i?( + + J F -l)= J R(X +F + -l)=. Lemma 7: Let { } be a sequence of natural numbers such that b n+ = b n+l +b n. Then {b n } has the following properties: 50 [FEB.

5 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS (i) R(b n -l)=r(b k -l) for all n>k if F k is the smallest Fibonacci number used in the Zeckendorf representation ofb k,k>, or if {b n } has b > b x and F k _ x < b - ^ < F k. (ii) R(b n -) = R(b n - l)r(f m ) -q,qn constant, 0 < q < R{b n -), where F m is the smallest Fibonacci number used in the Zeckendorf representation ofb n,m>; (iii) R(b n+ ) = R(b n ) +R(b n -) = T(b n \ n>k,ns in (i), where T(N) is the number of representations of JVas sums of Fibonacci numbers, where both F x and F can be used; (iv) R(b n+c) = R(bJ + cr(b n -l)=r(b n+c _ ) + R(b n -l\ n>k. Proof: Klarner [] used the Zeckendorf representation of b n to prove (i) for n sufficiently large; n > k as in the second statement appears in []. The proof of (ii) relies on Lemma 5 and mathematical induction. Take F n <b n <F n+l -. Let b n = F n +K, 0<K<F n _ x -. Assume part (ii) holds for all integers K = F n _ x. lfo<k< F n _ -, Lemma 5 and the inductive hypothesis give R(h )=R(K) + R(F n _ + K) = [R(K-l)R(FJ- gi ]HR(F n _ +K-l)R(F m )-q ] = [R(K-T)+R(F _ +K-l]R(F m )-( qi +q ) = R(F n +K-l)R(F m )-q = R(b n ~l)r(f m )-q, 0<q <R(b n -l\ since 0 < q x + q < R(K-) +i?(f _ + K -) = R(F n + K -) = R(b n -), again using the inductive hypothesis. A proof by induction can be made from each of the other two parts of Lemma 5, extendingkto the intervals F n <K< F _ -, and F n _ <K< F n _ x -, but is omitted here in the interest of brevity. To prove (iii), using (i) and (ii), R(b n+ ) = R(b n+ - l)r(f m+ ) -q = R(b n - l)(r(f m ) +) - q = (R(b n -l)r(f m )-q)+r(b n -l)=r(h ri )+R(h n -l). Next, take N = b n and use T(N)=R(N)+R(N-l) as in []. Note: The notation is not standardized; the meanings of R(N) and T(N) are reversed in [] from those used in this paper. Part (iv) follows from R(F n + c ) = R(F n ) + c, using (ii) to write where, also from (iii) and (i), = (R(b n - l)r(f m ) - q) +cr(b -l)=r(h ) +cr(b -), R(K + c) = R(K + c-) +R(b n+ c- - ) = R{K+c-) +R(K ~ ).. FORMULAS FOR R(N) BASED ON ZECKENDORF REPRESENTATION A formula for R(N) for whole sequences {bj, b n+ =b n+l +b n, can be written, or R(N) for large integers N based on the Zeckendorf representation of N, by repeatedly using Theorem, Lemmas and 6, and formulas for R(F n+p + N) as developed next. Let the largest Fibonacci number contained in Nhe F n ; equivalently, F n is the largest term in the Zeckendorf representation of N 9 and F n <N < F n+l -. To count the number of ways to represent N as sums of distinct 999] 5

6 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS Fibonacci numbers, first find the largest two Fibonacci numbers in AT and then apply formulas of the form R(F +p + N). Lemma 8: Let F < N < F +l -. Then R(F + + N) = R(N) + R(N-F ); R(F n++ N)==R(N) + R(F +l --Ny, R(F n+ + N) = R(N). Proof: Let M = N + F n+l in Lemm&,where F + <M <F + -. Then R(F +l +N) = R(F +l + N - F n+ ) + R(F n+l + N- F n+l ) - R(F + + N- F + ) = R(N-F ) + R(N)-R(N-F n+l ) = R(N-F ) + R(N), where R(N - F n+l ) = 0 because N < F n+l. Let M - N + F n+ in Lemma, where F n+ < M < F n+ - ; m + +N) = R(F n+ + N- F + ) + R(F n+ + N- F + ) - R(F + + N- F + ) = R(N) + R(N + F +i )-R(N-F n ) = R(N) + [R(N - F ) + R(N)] - R(N - F ) = R(N). Let M - N + F n+ in Theorem, where F n+ < M < F + - ; Theorem: Let F n <N<F n+l -. R(F n+ + N) = R{F n+ - - (F n+ + N)) + R((F n+ + N) - F n+ )) = R(F n+l --N) + R(N).V Then R(F +k+l + N) = (k + l)r(nxk>l, (9) R(F n+k +N) = kr(n) + R(F n+ --N), k>l (0) Proof: Assume that R(F n+j+l + N) = (j + l)r(n) holds for j < k; the case k - was established in Lemma 8. Consider By the first part of Lemma 5, R ( F n+(k+l)+l + # ) = R (F{n+c+\)+ + N \ n < F n+\ < F (n+k+)- m + k + + # ) = *tf»* + l + * ) + WW = (k + l)r(n) + R(N) = [(k + l) + l]r(n), establishing the formula for R(F n+k+l + N) by induction. The proof of the even case is similar, again taking the case k = from Lemma 8, and using Lemma 5; therefore, it is omitted here. Theorem can be used as a reduction formula to write R(N) for large N. For example, R(69) =R(F ) = R(97) + R(--97)= (9)+ 6 =, so i?(69) = since R(97) = 9 and R(5) - 6 are known from data. However, Theorem can be written in another form that is even more useful for computation, as given in Corollary.. 5 [FEB.

7 THE NUMBER OF REPRESENTATIONS OF FUSING DISTINCT FIBONACCI NUMBERS Corollary.: Let F n <N<F n+l -. Then R(F m + N) = R(F m _ n+l )R(N) + r, m-n> y where r = 0 if m-n is odd, and r = R(F n+l - - N) if m-n is even. Proof: The result follows from R(F n ) = [w/] for [x] the greatest integer in x from [] and []. Let /w-w = * + l, then m = n + k + l and [(/w-«+ l)/] = ifc + ; therefore, R(F m + N) = [(m-n + l)/]r(n)by(9). Similarly, let m-n = k in (0). D SPECIAL VALUES FOR R(F n ±K) We write some special formulas useful in breaking down expressions for R(N) by putting special values into equation () and Corollary.. Expressions for k = 0,, and in Lemma 9 appear in []. We also find integers m and q such that R{M) = mr(m-l)-q. Lemma 9Special values for R(F n -t±k): Let [x] be the greatest integer contained in x, and let 0 < * < F _ t. Then R(F n -l + k) = R(F n+l -l-k) has the following values, 0 < k < 8. n-\' * = 0: *(F -) = R(F _ -) =, n> * = : R(F n ) = R(F. -) = R(F ) =[n/], n> k = : R(F + \) =i?(f n+ -) =R(F n _ l ) =[(n-l)/], «> k = : R(F n + ) =R(F n+.-) = R(F n _ ) =[(/i-)/],»>5; k = : R(F + ) =R(F n+: -5) = «-, «>6: k = 5: R(F +) =R(F n+ -6) = fl(f _ ) =[(»-)/], n>6. k = 6: R(F n + S) =R(F n -7) = «-, «>7; k = T. R(F n + 6) =R(F n+ -8) =w-, n > 7 k = 8: R(F n + T) =R(F + -9) =i?(f _ ) =[(»-)/], n>l Lemma 0-Special values for R(F lc ±K) and R(F c+ ±K): Considering n even and n odd, R{F n ±K) has the following values: R(F c ) =R(F c+l ) =R(F c _ ) + \; R(F c + l) =R(F c _ l ) = R(F c _ ); R(F c+l + l) =R(F c ) =i?(f c+ ); R(F c+l +) =/?(F c _ I ) = ^ + ); R(F c+l -l) =R(F c -l) =- Lemma : Let A" be an integer whose Zeckendorf representation has F m + F k for its smallest two terms. If k - so thatkends with F m + l, m>a, then R(K) = R(K-l),m odd; i?( ) = R(K-\)-R(K-),m even; () If k = so that. ends in F m +, w > 5, then R(K) = R(K-\)-R(K-),m odd; R(K) = R(K-l),m even; () If K ends in F m + F c, c >, then i?(r) = /?( -) + R(K +); () IfKends in F m + F ch,c>, then i?( ) = R(K-) + i?(^ + ). (5) (ii) 999] 5

8 THE NUMBER OF REPRESENTATIONS OF JV USING DISTINCT FIBONACCI NUMBERS Proof: A proof can be written by induction following this outline. Calculate () for K = F c + lmdk - ^ic+i + - Equation () can also be calculated for K - F m + F c + and K F m + F c+ +. Then assume that () holds for all K such that K < F n _ l - and use (5) from Lemma 5, R(F n + K) = R{F n _ + K) + R(K), 0<K<F _ -l, calculating each part of (). Repeat for the other two parts of Lemma 5. () and (5) can be proved by substitution into () and (). When K ends in F m +F c, K+l ends in F m +F c + l, so replacing K by K + in () in the even case yields (). When K ends in F m + F c+l, then K + l ends in F m +F c+l +, which means that R(K +) = R(K) for the odd case of (). Also, K + ends in F m + F c+l +, which means that R(K + ) = R(K + l)-r(k-l) from the odd case of (). Puttingthese together gives (5). D Theorem : Let F m +F k be the smallest Fibonacci numbers in the Zeckendorf representation of M. Then R(Ad) = R(M-l)R(F k )-q, 0<q<R(M~l). (6) If the Zeckendorf representation of Mends in F c+l + or F c +, where c >, then q - 0. If M ends in F c+l +, q = R(M-); F c + l, q = R(M-). If m-k is odd, q = 0. If Mends in F w +F c, c>, then q = (c-l)r(m-l)-r(m + l); if Mends in F w+ +F c+l, c>, then q = (c-l)r(m-l)-r(m+). Proof: Apply Lemma 7(ii) and Lemma. When m-k is odd, q = 0 by Theorem. Corollary.: Let K be an integer whose Zeckendorf representation has smallest two terms F m +F k. Then R(K) = cr(k -) when k = c and m is odd, and when k = c + and w is even. 5. M(MF k ) AND R(ML k ) Below, R(MF k ) can be obtained by putting MF k into Zeckendorf form and then applying Theorem repeatedly. We list Zeckendorf representations of MF k for M < 8, taking smallest entry F k _ c > F and write R(MF k ) for M<9 = Lj. hf k = V* = L F t = L*F k = L 6 F k = F k lf k *F k = F k+ +F k +F k _ 5F t = F M +F k _,+F k _ 6F k -F k+i +F k+ i+f k _ F k = F M +F k _ A &F k = F k+ +F k +F k _ 9F k = F M +F M +F k _ +F k _ lof k = F k+ +F M +F k _ +F k, UF t F t lf k m \5F k \6F k HF k UF k -F k+i +F k _ = F k+ +F k _ = F M +F k+ +F k +F k^+f k _ = F M +F k _ l + F k _, + F k _ 6 = F k+5 + F k+l + F k _ + F k^ = F k+5 +F k+ +F k _ +F k _ 6 = F k+5 +F k+ +F k +F k _ + F k _ 6 - F M + F M + F k _ t + F k _ 6 = F ki. 5 + F M +F k+l +F k _ 6 = F k+6 +F k _ 6 = F k+j - F k _ = F M +F k -F k _ i = F M +F k +F k _ = F M +lf k +F _ A = F M +F t +F k _ = F k+ +F k +F k _ ~ ^k+5 +^k~ ^k-5 = F k+s +F k -F t^ = F M +F -F k _ i = F k+5 +F k -F^s = F k+s +5F k -F k _ 5 = F k+5 +6F k -F k _ s Lemma : For MF k such that L^^i <M<L lc + h k>c +, the smallest Fibonacci number in the Zeckendorf representation of MF k is F k _ c, and the largest is F k + c _ x or F k+c, depending upon the interval, where 5 [FEB.

9 THE NUMBER OF REPRESENTATIONS OF FUSING DISTINCT FIBONACCI NUMBERS <MF k <F k+c, c _j <M<L lc, F k+c < MF k < F k+ch, Li c <M< L C+h Proof: Lemma Is illustrated for M < 8. Assume it holds for all integers 0 < Q < L ^ ; i.e., the largest term in QF k is F k+c _ and the smallest is F k _ c _ when Z^_ < Q < L ^. Since LcF k =F k+c + F k _ c (see [6]), MF k = QF k + L^k = F k+e + QF k + F k c has largest term F k+c and smallest term F k _ c for L^ < M = Z^ + Q < L^c+l. The subscript difference between F k _ c _ and the next smallest Fibonacci number used in the Zeckendorf representation of MF k is even. For L e _ l <M<L C, since L ^ F * = F k+c _ t -F k _ c+l (see [6]), MF k = L ^ F * + QF k = F k+c _ t - + QF k,0<q<l c _. Assume the largest possible term in the Zeckendorf representation of QF k is F k+c _ and the smallest term is F k _ j for L^^T, <Q< LI C - There is no modification of terms for the Zeckendorf representation in adding F k+c -\, but the smallest term in the Zeckendorf representation of MF k becomes F k _ c for l^,^ <M < Z^ since Fk-n ~ F k - c+ i = {F k _ i - F k _ c+ ) + F k _ c = (Fk-y-i + F k - i F k _ c+ ) + F k _ c. Thus, the largest term is i^i +e -i an d t n e smallest is F k _ c for MF k, when L^^ < M < Z^. Note that the subscript difference between F k _ c and the next smallest Fibonacci number used in the Zeckendorf representation is odd. R(MF k ), < M < 9 = L,, k > c + for Smallest Term F A _ c R(F k ) = R(F k _ 0 ) R(F k ) = R(F k, ) R(F k ) = R{F kl )-l R(F k ) = R(F k _ )- = R(F k _ ) + l =L, R(SF k ) = 5R(F k _ ) R(6F k ) =5R(F k _ ) R(lF k ) = 5R(.F k _ )-l 7 = L R(ZF k ) = 8fl(F t _ )- R(9F t ) = 8/?(F t _ )- R(lOF t ) = 8K(F _ )-5 R(llF k ) = 5R(F k _ )- =5R(F k _ 6 ) + l = Z, R(lF k ) =0R(F k _ 6 ) R(lF k ) = lr(f k _ 6 ) R(UF k ) =lr(f k _ 6 ) R(lSF t ) =R(F^6) R(l6F k ) = lr(f k _ 6 ) R(\lF k ) =WR(F k _ 6 ) R(\*F k ) =7R(F k _ 6 )-l 8 = R(\9F k ) =5R(F k _ 6 )- R(0F k ) = 8fl(F*_ 6 )-6 R(lF k ) =LR(F t _ 6 )-8 R(F k ) =l6r(f k _ 6 )-l R(F k ) = 0/?(F t _ 6 )-0 R(F k ) =0R(F k6 )-W R(5F k ) = l6r(f k. 6 )-9 R(6F k ) =LR(F M )- R(UF k ) = 8R(F _«)- R(SF k ) =5R(F k _ 6 )-U R(9F k ) =7R(F k _ 6 )-6 =!R(F k _ s ) + l 9 = L, 999] 55

10 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS Theorem 5: When L c _ l <M< C+, k >c +, R(MF k ) = R(MF k -l)r(f k _ c )-q, (7) where R(MF k -l) = R(MF c+ -). Further, q = 0 for L c _ l KMKI^, while q = R(MF c+ -) Proof: The assertions follow from Theorem by taking k = c + in (7), since we have F k _ c as the smallest term of the Zeckendorf representation of MF k by Lemma. When L c _ < M <L lc, the last two terms in the Zeckendorf representation are F m +F k _ c, where (m- k + c) is odd; thus, in using Theorem repeatedly to evaluate R(MF k ) from its Zeckendorf representation, we will have q = 0 by Corollary.. When L C <M< L c+l, the subscripts of the last two terms will have an even difference, so a remainder term will be involved. Taking k - c + to give the smallest F m -F gives q = R(MF c+ - ) by Theorem in the interval where q ^ 0. Next, we note that the values R(MF c+ -) form palindromic subsequences such that: Also of interest, we have R((lc-i + K)F k -) = R((L lc -K)F k -), <K<[L e _ \; R((L c+ K)F k -l)) = R((L c+l -K)F k -i), 0<K<[L c _ l /]. R(L c F k -l) = R(L c+l F k -l); R(L c _ l F k -l) + = R(L c F k -l). Corollary 5.: R(L L p -l) = (p-l), n>p +, p>. Proof: Vajda [6] gives equation (7a), equivalent to J +, + L n- P = L n L P> P even, Since L n+p + L _ p = F n+p+l + F +p _ l +F n _ p+l +F _ p _ l, the smallest Fibonacci number used in the Zeckendorf representation is F _ p _i. Theorem gives R(L n+p + L. p ) = R(L n+p + L _ p - WiF^) = R(I^L p -l)r(f^-9. Since we only want R(L L p -l), we calculate R(L +p + L _ p -) when F n _ p _ x =F or, for n-p =, n = p +, so that R(L +p + L _ p -) has a constant value for n>p +. R(L +p + _ p -) = U(Z^ + - ) = i?(f /?+ +F /?+ +) = i?(f p++ ) + i?(f /?+ -5) - q = (/>-l) + (/>-) = (p-l), where we have applied earlier formulas from Theorem and special values for Thus, R(L L p -) = (p -) for/? even. R(F n+l -l-k). 56 [FEB.

11 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS Similarly, for p odd, +/?--p has F n _ p _ as the smallest Fibonacci number in its Zeckendorf representation. Again calculate R(Ln+ p - L^p -) for the smallest value for F n _ p _ =F, which occurs for n-p =, n = p +. Then Thus, R(L n L p -T) = (p-l) [i]. a RiL,^ - L -) = R(F p+5 + F p+ - 8) Corollary 5.: R(F p F -l) = F p, n>p, p>. = R(F p+ - 8) = (/7 - ) = (/> -). for p odd, establishing Corollary 5. and proving Conjecture of Proof: F c+i F k and F c+ F k both have -c a s ^ e smallest term in the Zeckendorf representation. Thus, Wienie) = W c+ if k ~ l)r(f k. c ) - q; W M F k ) = R(F c+ F k - l)r(f k _ c ) - q. When k > c +, R(MF k -) has a constant value. When k = c +, while R(F c +ifk -) = R(F c+ if c+ -) = F c+ i?(c+ -) = Wic + F c+ -) = c +, applying two identities from Carlitz []. Thus, R(F p F n -l) = F p, establishing Corollary 5. and making a second proof of Theorem in []. The Lucas case R{ML k ) is very similar, relying on [6] for =F k+p +F k _ p, p odd, and = F k+p -F k _ p, p even. When F c _ <M<F c, the smallest term in the Zeckendorf representation of ML k is F k _c+i, w h i l e t h e largest is +c-> F c _ <M<F c _ u or +c _i, c-i ^ M<F c,fc>c + l. ~ F L k = F L k = F 5 L k = Zeckendorf Representations for ML k9 < M < = ++- = ++- = + + F k+l + F k-l + F k- = =+5+-5 = = + ~ - 6 = i + - i = = 8 = i + - i = = = = F k + l = _ + _ 7 = = ] 57

12 THE NUMBER OF REPRESENTATIONS OFWUSING DISTINCT FIBONACCI NUMBERS R(ML k ), < M < = F 8, k > c + for Smallest Term F A _ c _j R(L k )- R(L k )-- R(L k ): i?(): ^(5) = R(6L k )-- ROh)- R(SL k ). R(9L k ). R(\0L k ) R(UL k ): i? ( _, ) - l i? ( _ ) - l R(F k _ )- 6 t f ( _ 5 ) - l UR(F k _ 5 )-5 lr(f k _ 5 )-7 6i?(_ 5 )-5 8i?(- 7 ) l6r(f k _ 7 ) l6r(f k _ 7 ) i?(): i?() = R(UL k )-- R(\5L k )-. i?(6): R{\lL k ) - R(\SL k )-- R(9L k ) R(0L k )-. R{\L k ). lsr(f k -7) 8 i? ( - 7 ) - l R(F k _ 7 )-7 0R(F k _ 7 )-U 0R(F k _ 7 )-9 R(F k _ 7 )-\6 0i?(_ 7 )-ll 0i?(_ 7 )-9 R(F k _ 7 )-l7 SR(F k _ 7 )-7 Theorem 6: When F c _ < M < F lc, k > c +, R(ML k ) = RiML, - l)r(f k _ c+ ) - q, (8) where R{ML k - ) = ^(A/Lje+i-); further, q = 0 for C _ < M < C _ ; and? = R(ML c+l ~) when c - i <M<F c. The proof of Theorem 6 depends on Theorem, and being similar to the proof of Theorem 5 is omitted here. We note that the values R(ML c+l -) form palindromic subsequences such that R((F c _ +K)L k -l) = R((F c -K)L k -l\ 0 < K <[F c _ ']; R((F c +K)L k -l)^r((f c -K)L k -i), < X <[F c _, ]. Theorem 7: R(F m ±F k ) and R(F m ±F k -l) Proof: By Corollary., 6. /?(F m ±F A ) have the following values: R(F m +F k ) =R(F m _ k+ )R(F k ), (m-k) odd; R(F m +F k ) = R(F m _ k+ )R(F k )-l, (m-k) even; R(F m - ) = R(F m _ k+l )R(F k _ ) +, (w - *) even; i? (, - ) = R(F m _ M )R(F k _ l ), ( «- *) odd; i?(,h--l)=i?(,., + ); i?(,--i)=i?( _ ft+ ). R(F m +F k ) = R(F m _ M )R(F k )+r. If m-k is odd, r = 0, and R(F m _ k+ ) = R(F m _ k+ ), making R(F m +F k ) = R(F m _ k+ )R(F k ). If M - * is even, r = /? ( F t F t ) = /?(F t _ ) = /? ( / ) -, and i?(,_ fc+ ) + l = i?(,_* + ), making i?(, + ) = i?(,_, + ) J R()-i. Equations (7) give R(F m ±F k -l) by examining the difference of the subscripts; note that the results for R(F m + F k ) agree with Theorem. Using Theorem, followed by Corollary., while noting that the greatest Fibonacci number in F k - is F k _ u R(F m -F k ) = R(F m l + (F k -)) = R(F m _ k+l )R(F k - ) + r. 58 [FEB.

13 THE NUMBER OF REPRESENTATIONS OF N USING DISTINCT FIBONACCI NUMBERS Note that R(F k - ) = RiF^). If m - k is odd, r = 0, while if m - k is even, r=r(f k --(F k -)) = R(0) = l D Corollary 7.: R{F r L t -) can be written as (i) R(F L p -i) = R(F p ) + l,n>p +,p>l; (ii) R(L n F p -) = R(F p+l ), n>p + \,p>. Proof: Vajda [6] gives equation (5a), equivalent to \ F n +P + K-p = F n L p> P e v e n > [F + p ~ F n - P = F n L p, P odd, By Theorem 7, R(F +p +F p -l) = R(F p+ ) = p + l, while R(F n+p -F _ p -l) = R(F p ) = p. So R(F n L p -\) = p + \,p even, and R(F L p -\) = p,p odd, which makes R(F L p -) = [p ] +, proving part (i) as well as Conjecture of []. Since [6] also gives [F n+p +Fn-p = L n F p, podd, [F +p -F n _ p = L n F p, pevm. in the same way, we can show that R(L n F p -l) = p + l, p odd, and R(L n F p -\) = p, p even, which can be rewritten in the form of (ii). Thus, we have proved part (ii) as well as Conjecture of[l]. Corollary 7.: Let F <N<F +l -. (i) RiL^) = R{F p )- = R{V.) +, p > ; (ii) R(L n+p + N) = R{F n + p _ x + N) + R(F n+p _ + N) = R(L p+l )R(N) + r, where r = 0 ifp is odd, and r = R(F n+l --N) if/? is even; (iii) R(L n+p -K) = R(F +p _ + (K-)\<K<F +p _. Proof: Since L p+l =F p+ + F p, let m = p + and k - p in Theorem 7 to write (i). Apply equation (0) to R(F n+p+l + i^+p _i + N) followed by Theorem to write the first part of (ii). Then use Corollary. and (i) to simplify, finally obtaining (ii). When <K<F n+p _, the largest term in the Zeckendorf representation of F n+p _ x -K is F n+p -- Then R(L n+p -K)= RiF^ + (F +p _, - Kj) = R(F n+p _ i -K) = R(F n+p _ - + K).U Corollary 7.: R(L +p + L _ p ) = (p - )R(L n _ p ) -l = (p-)/?^^,) - (p -); R(L n+p -L n _ p ) = (p-l)r(f _ p _ ), n-p>. Proof: Let N = L _ p = F _ p+l +F n _ p _ x in Corollary 7.. Then R(F n+p _ l + N) + R(F +p. + N) = {p- l)r(l _ p ) + (p- )R(L _ p ) + R(F _ p+ - - F _ p+l - F n _ p _ x ) 999] 59

14 THE NUMBER OF REPRESENTATIONS OF JV USING DISTINCT FIBONACCI NUMBERS = (p - )R(L _ p ) + R(F _ p _ ) = (lp - )R(L _ p ) + R(L _ p ) - = (/7 - )R{L n _ p ) -l = (p- )[R(F, p _ l ) -] - ^(p-l)r(f n _ p _ )-(p-l). Now let K = L n _ p in Corollary 7.. Then finishing Corollary 7.. R (L n+p - L _ p ) - R(F +p _ +F _ p+l +F _ p _ l -) = (p-l)r(f p+l + F n _ p _ -) = (p-l)(r(f _ p _ l -)) = {p-\)r(f n _ p _ ), Corollary 7: R(L n L p -) = (p-), n>p +, p>. Proof: Vajda [6] gives L n+p + L n _ p - L n L p when/? is even, and L n+p - L n _ p = L n L p when/? is odd. The smallest Fibonacci numbers in the Zeckendorf representations are F n _ p _ x and F n _ p _, respectively. Since also R(L n+p ±L n _ p -l) = R(L n L p -l), apply Theorem to Corollary 7.. This also proves Conjecture in []. Corollary 75: R(5F n F p -l) = (p-l), n>p +, p>. Proof: L n+p + L n _ p = 5F n F p, p odd; L n+p - L n _ p = 5F n F p, p even, also appear in [6], giving an easy identity as in Corollary 7.. REFERENCES. Marjorie Bicknell-Johnson. "A Note on a Representation Conjecture by Hoggatt." In Applications of Fibonacci Numbers 7 (to be published by Kluwer Academic Publishers, 998).. Marjorie Bicknell-Johnson & David A. Englund. "Greatest Integer Identities for Generalized Fibonacci Sequences {H n }, H n = H n _ x +H _ " The Fibonacci Quarterly. (995): L. Carlitz. "Fibonacci Representations." The Fibonacci Quarterly 6A (968): David A. Klamer. "Partitions of Ninto Distinct Fibonacci Numbers." The Fibonacci Quarterly 6A (968): David A. Klamer. "Representations of N as a Sum of Distinct Elements from Special Sequences." The Fibonacci Quarterly (966): S. Vajda. Fibonacci & Lucas Numbers, and the Golden Section. Chichester, England: Ellis HorwoodLtd., 989. AMS Classification numbers: B9, B7, Y55 >!! 6 0 [FEB.