CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ t + G n _ c FROM MATRICES WITH CONSTANT VALUED DETERMINANTS Marjorle Bicknell-Johnson 665 Fairlane Avenue, Santa Clara, CA 955 Colin Paul Spears University of Southern California, Cancer Research Laboratory, Room 26 33 North Mission Road, Los Angeles, CA 933 (Submitted June 994) The generalized Fibonacci numbers {G w }, G n = G n _ x + G n _ c, n>c,g =,G l = G 2 =--= G C _ X =, are the sums of elements found on successive diagonals of Pascal's triangle written in leftjustified form, by beginning in the left-most column and moving up (c-) and right throughout the array []. Of course, G n =F n, the n th Fibonacci number, when c-2. Also, G n = u(n-l; c-,), where u(n; p, q) are the generalized Fibonacci numbers of Harris and Styles [2]. In this paper, elementary matrix operations make simple derivations of entire classes of identities for such generalized Fibonacci numbers, and for the Fibonacci numbers themselves. Begin with the sequence {G n }, such that. INTRODUCTION G = G n _ l+ G _ 3, n>3, G =, G x = G 2 =. (.) For the reader's convenience, the first values are listed below: n G 2 3 4 2 5 3 6 4 7 6 8 9 9 3 9 n 2 3 4 5 6 7 8 9 2 2 G n 28 4 6 88 29 89 277 46 595 872 278 These numbers can be generated by a simple scheme from an array which has,, in the first column., and which is formed by taking each successive element as the sum of the element above and the element to the left in the array, except that in the case of an element in the first row we use the last term in the preceding column and the element to the left: 4 3 [4]i 29 2 6 [9]-* 6 89 3 9 28 88 277 (.2) If we choose a 3 x 3 array from any three consecutive columns, the determinant is. If any 3x4 array is chosen with 4 consecutive columns, and row reduced by elementary matrix methods, the solution is 996] 2
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ x + G n _ c (l o We note that, in any row, any four consecutive elements d, e,f, and g are related by ) -3 4, (.3) d~3e + 4f = g. (.4) Each element in the third row is one more than the sum of the 3k elements in the k preceding columns; i.e., 9 = (3+ 2 + + + + ) +!. Each element in the second row satisfies a "column property" as the sum of the three elements in the preceding column; i.e., 6 = 28 + 9 + 3 or, alternately, a "row property" as each element in the second row is one more than the sum of the element above and all other elements in the first row; i.e., 6 = (4 +3 + 4 + + ) +. Each element in the third row is the sum of the element above and all other elements to the left in the second row; i.e., 28 = 9 + 6 + 2 +. It can be proved by induction that which compare with G + G 2 -fg 3 + - + G = G B+3 -l, G +G 4 +G 7 + - G 2 + G 5 +G i + - F l+ F 2 +F 3 + - +F=R F 2 + ^ 4 + ^ 6 + ' +G 3k = G 3 * + r, +G 3k+l ' + * Jr 3k+2 ' + * i 2k+l ' +F 2k «+2 J 3k+2> ~( J 3k+3> ~ * 2k+2> = F 2k+l ~> (.5) (.6) (.7) (.8) (.9) (.) (.) for Fibonacci numbers. The reader should note that forming a two-rowed array analogous to (.2) by taking, in the first column yields Fibonacci numbers, while taking,,,..., with an infinite number of rows, forms Pascal's triangle in rectangular form, bordered on the top by a row of zeros. We also note that all of these sequences could be generated by taking the first column as all l's or as, 2, 3,..., or as the appropriate number of consecutive terms in the sequence. They all satisfy "row properties" and "column properties." The determinant and matrix properties observed in (.2) and (.3) lead to entire classes of identities in the next section. 2. IDENTITIES FOR THE FIBONACCI NUMBERS AND FOR THE CASE c = 3 Write a 3 x 3 matrix A = {a tj ) by writing three consecutive terms of {G n } in each column and taking a n = G n, where c = 3 as in (.): 4,= \Gn+2 a n+p J n+p+\ ^n+p+2 G, n+q J n+q+\ ^n+q+lj We can form matrix A n+l by applying (.), replacing row by (row + row 3) in A followed by two row exchanges, so that (2.) 22 [MAY
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G -G, + G n n \ n det 4, = det^+. Let n = l,p = \q = 2 in (2.) and find det^ = - l. Thus, det 4,: - for (2.2) 4,= (G n w+l \G n +2 G +\ &n+2 G,<, G J n+3 n+4j (2.3) As another special case of (2.), use 9 consecutive elements of {G n } to write 4 = a (G G +l G. n+3 3*^ J n+5 G w+7 (2.4) which has det ^ =. These simple observations allow us to write many identities for {G } effortlessly. W e illustrate our procedure with an example. Suppose we want an identity of the form ag n + G n+l + yg n+2 = G +4. W e write an augmented matrix A, where each column contains three consecutive elements of {GJ and where the first row contains G, G w+, G w+2, and G + 4 : 4T = ( G n G n+ i Kpn+2 G n+i G n+2 G +3 G n+2 G +3 G n+ 4 G n+4 ] G n+5 G n+ 6; Then take a convenient value for n, say n = l, and use elementary row operations on the augmented matrix A*, (I 2>\ (\ A A; = 2 4 2 3 6 -> to obtain a generalization of the "column property" of the introduction, G + G n+l + G +2 = G +4, which holds for any n. While w e are using matrix methods to solve the system (2.5) ag + /?G + + yg n+2 = G n + 4, ag n+ i + PG n+ 2 + yg +3 = G t «+5> a G + 2 +PG n + z +YG n+a = G, n+6> notice that each determinant that would be used in a solution by Cramer's rule is of the form det 4? = det A n+l from (2.) and (2.2), and, moreover, the determinant of coefficients equals - so that there will be integral solutions. Alternately, by (.), notice that ( a, /?, y) will be a solution of ag n+3 + /?G +4 + yg n+5 = G n+ whenever ( a, /?, y) is a solution of the system above for any n > so that w e solve all such equations whenever w e have a solution for any three consecutive values of n. 996] 23
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ x + G n _ c We could make one identity at a time by augmenting 4, with a fourth column beginning with G n+w for any pleasing value for w, except that w < would force extension of {G n } to negative subscripts. However, it is not difficult to solve K (G J n+\ C C C ^ KJ n+l ^ 7 + 2 KJ n+w J n+2 G,* ^n+3 G n+w+l V^w+2 ^n+3 Sn+4 ^n+w+2; by taking n = and elementary row reduction, since G w+2 - G w+i = G w _ x by (.), and so that 4= ( G w G, 2 G, w+ w+2y -» ( l G w l o o G w-2 G ^ -> (I G w 2^ G G w-3 w-lj G n+w G w _ 2 Cr n + G w _ 3 G n+ + G^^G,^ For the Fibonacci numbers, we can use the matrix 4*> A = F * n AH- F M n+q ^n+q+l (2.6) for which deta n = (-l)dgta n+. Of course, when q-\ det4,=(-l) where, also, d e t ^ F n F n+2~ F n+i> i v i n g t h e well-known Solve the augmented matrix A% as before, by taking n = -l, (-ir + =F n F n HI n+l -F 2 n+2 M n+l ^n+l ^n+w F F / ' r n+2 r n+w+l J (2.7) to obtain of - ' o i F 77 77 I 77 77 _ 77 w- «w «+l «+ w (2.8) Identities of the type ag n +/3G n+2 +yg n+a = G +6 can be obtained as before by row reduction (G < = G n+l Kpn+2 G n+2 G n+3 G n+4 G n+4 G n+5 G n+6 G n+6 ) G n+ G n+$j If we take n =, det A$ =, and we find a =, j5 = 2, y -, so that In a similar manner, we can derive G n+6 = G n + 2G n+2 + G n+4 G «+9 = G W ~ 3 G «+3 + 4 G W 6, (2.9) (2.) 24 [MAY
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ { + G _ where we compare (.4) and (2.). For the Fibonacci numbers, solve by taking n = l, so that Similarly, ^w+2 ~~ G 2G n+4 + 5G +8, K- K, A?+2 -^n+4 Ai+ -Tfi+3 ^w+5 A; = 2 5 3 -» - Ai+4 _ r n + 3r n+2. Ai+8 = Ai + 'Ai+4- In the Fibonacci case, we can solve directly for i^+ 2 p f rom (2.) (2.2) (2.3) (2.4) by taking w = -, _/" =! " " +/ ' F» n+2p -*w+l ^n+p+l -^n+2p+lj (\ F F ^ J -Tp-l ^2/7- v ^ i 7, 2 P ; -> f ( F ^ - F ^ F,, ) / ^ 2p' - p j -> ( - ) ^ since F p F 2 _ -F /? _ i^/7 = (-l) p l F p mdf 2p = F p L p are known identities for the Fibonacci and Lucas numbers. Thus, K + 2 P = (-V p - F + L p F +p. (2.5) Returning to (2.9), we can derive identities of the form «G +/?G +2 + yg n+4 = G +2w from (2.) with p = 2, # = 4, taking the augmented matrix A with first row containing G n, G +2, G +4, It is computationally advantageous to take n = -l; notice that we can define G_ x -. We J n+2w' make use of G 2w - G 2w _ x - G 2w _ 3 from (.) to solve obtaining < = 2 a 3 G 2w -> 2w+iy G W 2w-l G 2w _^ G lvt -G lvi _ x -> G \ G 2w+ - G 2wJ G 2w-3 f G -> G 2w-l ^ 2 ^ - 3 2w-3 -> G 2w _ 2 -G 2w _ 3j 2w-27 /> J G \ 2w-5 G 2 M M - G 2 W _ 3 G 2w _ 3 G n+ 2w ~ G 2w _ 5 G n + (G 2w _ G 2w _ 3 )Cj n+2 + G 2w _ 3 G ; n+4- (2.6) 996] 25
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ x + G n _ we have In the Fibonacci case, taking «= -, < = 77 E" r n+2 r n+2w ) -*w+l Ai+3 ^n+2w+lj A H - 2 w ->4!,= F 2 K 2w-l 2w -*2w-2^n ~^~ ^2 w^n+2 Returning to (2.6) and (2.6), the same procedure leads to G n+3w ~ G 3w-6 G n The Fibonacci case, derived by taking n = -, K = + \ G 3w ~~ 4 & 3 w _ 3 ) & w + 3 + G 3w-3 G n+6 (F ^n+3 ^n+3w V «+l ^n+a ^n+3w+lj - > n -/s K 2w-2 2w (2.7) (2.8) gives us A?+3w ~ A?^3w-3 '2+ A?+3^3w ' A (2.9) where F 3m /2 happens to be an integer for any m. Note that d e t ^ =(-) 2, and hence, det A n ^ ±- We cannot make a pleasing identity of the form ag n + j3g n+4 + yg n+s = G w+4w for arbitrary w because det 4? * ±, leading to nonintegral solutions. However, we can find an identity for {G n } analogous to (2.5). We solve ag_ x +/3G p _ x + yg 2p _ x = G 3p _ ly ag +/3G p +yg 2p = G 3p, ag x + pg p+l + yg 2p+l = G 3p+l for (a,j3,y) by Cramer's rule. Note that the determinant of coefficients D is given by D = G 2p G p _ l - G p G 2p _ x. Then a-aid, where A is the determinant G 3p _ x G p _ a 2/?-l G 3p+ G p+\ G 2p+l After making two column exchanges in A, we see from (2.) and (2.2) that A - D, so a =. Then J3 = BID, where B is the determinant B- G 3p-\ G 2p- G *p J 2P G 3p+\ G 2p+ G 2p G 3p-\ ~ G 3p G 2p- Similarly, y -CID, where C = G fi p^.\ - G P G 3 P ~\ Thus, G w + 3 p = G n + G n+p( G 3p-l G 2p ~ G 3 P G 2p-l) I D + G n+2p( G 3p G p-l ~ G 3p-\ G p) < A where D= ( G 2 P G p -\-G p G 2p -\). The coefficients ofg n+p and G n+2p are integers for p -,2,...,9, and it is conjectured that they are always integers. 26 [MAY
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ l 4- G _ As an observation before going to the general case, notice that identities such as (2.9), (2.), and (2.) generate more matrices with constant valued determinants. For example, (2.9) leads to matrix B n, (G ^n+p B» = where deti? = det +2. G n+2 ^Gn+4 The general case for {G n } is defined by G n+p+2 ^n+p+4 r \ ^n+q G n+q+2 ^n+q+4) 3. THE GENERAL CASE: G n = G n _ t + G n _ e G = G l _ + G^c,/i>c,whereG = O s G = G 2 = - = G c. = l. (3.) To write the elements of {G n } simply, use an array of c rows with the first column containing followed by (c-) l's, noting that, 2, 3,..., c will appear in the second column, analogous to the array of (.2). Take each term to be the sum of the term above and the term to the left, where we drop below for elements in the first row as before. Any cxc array formed from any c consecutive columns will have a determinant value of ±. Each element in the c th row is one more than the sum of the ck elements in the k preceding columns, i.e., which can be proved by induction. It is also true that G + G 2 + G 3 + - + G c t = G c(t+) -l, (3.2) Q + Ga+Ga +. -. + G ^ G ^ - l. (3.3) Each array satisfies the "column property" of (2.5) in that each element in the (c- l) st row is the sum of the c elements in the preceding column and, more generally, for any n, G n+c _ 2 = G n _ c + G n _ {c _ X) + + G n _ 2 + G n _ x (c terms). (3.4) Each array has "row properties" such that each element in the z th row, 3 </' <c, is the sum of the element above and all other elements to the left in the (/* -)** row, while each element in the second row is one more than the sum of the elements above and to the left in the first row, or G + G c + G 2c + G 3c + -. + G d k =G d t + -l, (3.5) G m + G c+m +G 2c+m + " ' + G ck+m = G ck+m+l> m = l > 2,..., C -, ( 3. 6 ) for a total of c related identities reminiscent of (.6), (.7), and (.8). The matrix properties of Section 2 also extend to the general case. Form the cxc matrix A nc - (a t j), where each column contains c consecutive elements of {G n } and a n = G n. Then, as in the case c = 3, d e t ^ ^ ^ i r M e t ^,, (3.7) since each column satisfies G n+c - G n+c _ { + G n. We can form 47+, c fr m Ai, c Y replacing row by (row + row c) followed by (c-) row exchanges. When we take the special case in which the first row of A n^ c contains c consecutive elements of {G }, then ^ c = ±. The easiest way to justify this result is to observe that (3.) cam be used 996] 27
CLASSES OF IDENTITIES FOR THE GENERALIZED FIBONACCI NUMBERS G n = G n _ x + G n _ c to extend {G } to negative subscripts. In fact, in the sequence {G n } extended by recursion (3.), G l = l and G x is followed by (c-) l's and preceded by (c-) O's. If we write the first row of A ntc as G w, G n _ h G _ 2,..., G _^^, then, for n =, the first row is,,,...,. If each column contains c consecutive increasing terms of {G }, then G n appears on the main diagonal in every row. Thus, A lc has l's everywhere on the main diagonal with O's everywhere above, so that det A lc =. That AoXA nc = ± is significant, however, because it indicates that we can write identities following the same procedures as for c - 3, expecting integral results when solving systems as before. Note that d e t ^ c = ±l if the first row contains c consecutive elements of {G }, but order does not matter. Also, we have the interesting special case that AttA nc - -* whenever A n c contains c 2 consecutive terms of {G }, taken in either increasing or decreasing order, c > 2. Det A n c = only if two elements in row are equal, since any c consecutive germs of G n are relatively prime [2]. Again, solving an augmented matrix A^ c will make identities of the form G n+w = a G n + a fin+l + a 2 G n+2 + "'+ <Xc-l G n+c-l for different fixed values of c, or other classes of identities of your choosing. As examples, we have: c=3 ^n+w ~ G n G + w-2 G n+\ G + w-3 G n+2 G w-l> c = 4 G n+w = G n G w _ 2 + G W+ G W _ 4 + G n+2 G W _ 5 + G +3 G W _ 2, c ~^ G n+w ~ G n G w-4+gn+fiw-s + G n+2 G w-6 + G n+3 G W _ 7 + G +4 G W _ 3, C C G n+w - G n G w-c+l + G n+l G w-c + G n+2 G w-c-l ^ " G n+c-l G w-c+2> c 2 A?+3 ~ Ai + r n+l + r n+ i, c = 3 G + 4 = G + G + + G w+2, c = 4 G w+5 = G + G w+ + G w+3, c = 5 G n+6 = G + G n+ + G +4, C = C G n+c+l = G n +G n+l +G n+c-v So many identities, so little time! REFERENCES. Marjorie Bicknell. "A Primer for the Fibonacci Numbers: Part VIII: Sequences of Sums from Pascal's Triangle." The Fibonacci Quarterly 9. (97):74-8. 2. V. C. Harris & Carolyn C. Styles. "A Generalization of Fibonacci Numbers." The Fibonacci Quarterly 2.4 (964):277-89. AMS Classification Numbers: B65, B39, C2 28 [MAY