1. INTRODUCTION. P r e s e n t e d h e r e is a generalization of Fibonacci numbers which is intimately connected with the arithmetic triangle.

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1 A GENERALIZATION OF FIBONACCI NUMBERS V.C. HARRIS ad CAROLYN C. STYLES Sa Diego State College ad Sa Diego Mesa College, Sa Diego, Califoria 1. INTRODUCTION P r e s e t e d h e r e is a geeralizatio of Fiboacci umbers which is itimately coected with the arithmetic triagle. beyod ad falls short of other geeralizatios. I sectio 2 the umbers a r e defied ad deoted by u(; p, q) where iteger ad q is a positive iteger. show to be (1. 1) x P (x - l ) q - 1 = 0. It at oce goes p is a o-egative The c h a r a c t e r i s t i c equatio is The umbers a r e r e p r e s e t e d i the usual maer i t e r m s of powers of roots of the equatio ad certai iitial coditios. I sectio 3 c e r - tai sums ad properties ivolvig sums a r e developed ad i sectio 4 there is made a begiig i the study of divisibility p r o p e r t i e s. The geeralizatio made h e r e may be compared with c h a r a c t e r - istic equatios obtaied i other geeralizatios: by Dickiso [2 J, x - x - 1 = 0 (a, c itegers) by Miles [4], x - x x - 1 = 0 (k itegral, ^ 2) r- -] r "t" 1 r >> by Raab [5 J, x - ax - b = 0 (a, b real; r itegral, ^ 1) by Feiberg [-8], x +1-2 x = 0, various positive ite gral values of u,. Geeralizatios by Basi [l] ad Horadam [3J ivolve alterig oly the iitial coditios of the Fiboacci sequece. The umbers studied here a r e special cases of sums defied i Netto _6J ad Dickiso [2J ad their defiitio ad relatio to the a r i t h - metic triagle appear i Hochster _7j. 2. THE NUMBERS u(; p, q) Let p ad q be itegers with p = 0 ad q > 0. The by defiitio the -th geeralized Fiboacci umber of step p, q is 277

2 278 A GENERALIZATION OF FIBONACCI NUMBERS December L'p+qJ (2. 1) u(; p, q) = 1 fr 7 ^ P \, > 1, u(0; p, q) = 1 Here [x] deotes the greatest iteger z x. I particular, u ( - l ; 1, 1) = f (the -th Fiboacci umber), = 1, 2,... u(; 0, 1) = 2 Whe the defiitio is related to the arithmetic triagle oe sees that u(; p, q) is the sum of the t e r m i the first colum ad the -th (coutig the top row as the z e r o - t h row) ad the t e r m s obtaied s t a r t - ig from this t e r m by takig steps p, q -- that is, p uits up ad q uits to the right. the It follows that u(0; p, q) = u(l; p, q) =... = u(p+q-l; p, q) = 1, u(p+q; p, q) = 2 If V is the backward differece operator, so that Vf(x) = f(x)-f(x-l), (2. 2) y q u(; p, q) = u(-p-q; p, q), > p + q. F r o m p r o p e r t i e s of biomial coefficiets ad it follows that y q u(; p, q) = y q ~ y u ( 5 P> <l) [-p-ql L p+q J V q u(;p, q,=" 2 ( - p 7 j - i p ) row = u( - p - q; p, q), ^ p + q.

3 1964 A G E N E R A L I Z A T I O N O F F I B O N A C C I N U M B E R S 27 9 T h i s p r o v e s (2. 2). I t e r m s of f o r w a r d d i f f e r e c e s t h i s is y u ( - q ; p, q) = u ( - p - q ; p, q), > p + q. T h e c h a r a c t e r i s t i c e q u a t i o a d i i t i a l c o d i t i o s c o s e q u e t l y a r e (2. 3) x P ( x - l ) q -. 1 = 0 Let u(; p, q) = 1, = 0, 1,,.., p 4- q- 1. p+q u(; p, q) = S c. x. r 1 1 w h e r e x., i = 1, 2,..., (p+q) a r e the r o o t s of (2. 3). T h e d e r i v a t i v e of f(x) = x P ( x - l ) q - 1 i s f'(x) = p x P - ^ x - l ) q x P ( x - l ) q - 1 = x P " 1 ( x - l ) q " 1 ( ( p + q ) x - p ). S i c e o r o o t of f'(x) is a r o o t of f(x), it follows t h a t f(x) h a s o m u l t i p l e r o o t. p+q H e c e the d e t e r m i a t of the c o e f f i c i e t s of c. x i = u(; p, q) = 1, = 0,..., p+q - 1 i s d i f f e r e t f r o m z e r o. T h e s y s t e m c a be s o l v e d by C r a m e r ' s r u l e w i t h V a d e r m o d i a s (as i s e v e r a l of the r e f e r e c e s ). It r e s u l t s t h a t c i = l / ( ( p + q ) x i - p) ad p+q +1 (2. 4) u(; p, q) = r X x. (p+q/x." -~p" ' r ^ ' l j 2 ' ' * ' T h e r e i s a p o s i t i v e r e a l r o o t x, > 1. T h i s follows f r o m f(l) < 0 a d f(2) = 0. Sice f'(x) 4 0 for x > 1 t h e r e i s o o t h e r r e a l r o o t > 1. A l s o x, I e x c e e d s the a b s o l u t e v a l u e of e a c h o t h e r r o o t. F o r if x- ^ x, i s a r o o t ad x~ ] ~ x, t h e

4 280 A G E N E R A L I Z A T I O N O F F I B O N A C C I N U M B E R S D e c e m b e r * P ( x 2 - l ) «= x 2 P x 2 - l q > x j P x r l ^ > 1 s o t h a t ( 2. 2 ) c a o t be s a t i s f i e d, a c o t r a d i c t i o. F r o m t h i s it f o llows l i m ( 2. 5 ) u < " + 1 : P» q > = X l x ' ~> co u ( ; p, q) 1 To s h o w t h i s, m e r e l y o t e i / 11 \ T u(+l; p, q)/x 1»> co u(; p, q) ->(X> " x / +2 ~ 1 r ^ u(; p, q ) / x 1 We r e m a r k t h a t if we c h o o s e i i t i a l c o d i t i o s u(0; p, q) = u ( l ; p, q) =... = u ( p + q - 2 ; p, q) = 1, u ( p + q - l ; p, q) = p + q + 1, t h e we h a v e a s e q u e c e (w(; p, q)) s w h e r e p+q w(; p, q) = 2 x_., = 0, 1, 2,... M o r e o v e r, a c o v e i e t f o r m for e x p r e s s i g u(, p, q) a r i s e s f r o m w r i t i g t h e d i f f e r e c e e q u a t i o a s (2. 6) u(; p, q) = (^) u ( - l ; p, q) - (^) u ( - 2 ; p, q) ( - l ) q ~ u ( - q ; p, q) + u ( - p - q ; p, q), ^ p + q. 3. SUMS T h e o r e m The r e l a t i o q - 1 ( 3. 1) 1 u(i; p, q) = 1 ( - 1 ) 1 ( q : l ) u ( + P + q - i ; P, q) - * l q h o l d s, w h e r e 8 i s K r o e e k e r ' s 5 ad (. ) = 1 i t h e c a s x lq l ' q = 1, i = 0. If (3 0 1) h o l d s for, for q > 2, t h e

5 1964 A GENERALIZATION OF FIBONACCI NUMBERS u(i; p, q) = u(+l; p, q) + u(i; p, q) q = 2 (-1) 1 ( q {) u(+l+p+q-i; p, q) q (-1) 1 ( q \ l ) u(+p+q-i; p, q) - l q q-1 = 2 ( - I ) ' ( q \ l ) u(+l+p+q-i; p, q ) - g Hece (3. 1) holds for + 1. Whe = 0, with q > 2, the (3. 1) b e- comes 0 q-1 2 U(i; p, q) = 2 (-1) 1 ( q ". 1 )u(p+q-i; p, q) - 8 lq q-1 = u(p+q; p, q) + 2 (-1) 1 ( q \ l ) = 1 = u(0; p, q) To complete the proof, we cosider q = 1. The u(i; p, 1) = u(p+l+i; p, 1) - u(p+i; p, 1) Hece 2 u(i; p, 1) = u(+p+l; p, 1) - u(p; p, 1] = u(+p+l; p, 1) - 8 ll.

6 282 A GENERALIZATION OF FIBONACCI NUMBERS December Theorem m (3.2) X ( - l ^ u t i j p. q ) q-1 k S 2(-l) k ( q )u(m+p+q-k;p,q) l-(-l)p + q 2 q lk=0 j=0 m+p + ( - D m + P + q 2 q S ( - l ^ u ^ p. q ) i=m+l + (-1) m " 1 2 q " 1 + ( - l ) ^ ^ " 1 2 q where Proof. = 0, p+q eve, ad *" = 1, p+q odd. Writig (-1) J u(m-j; p,q) = (-1 ) m ~ J u(m+p+q-j; p, q) + ( - l ) m ' j _ 1 ( q ) u(m+p+q-j~l; p, q) ( - l ) m + q ( q ) u(m+p-j; p,q) ad summig for j = 0, 1,..., m gives for the sum S, q-1 k m-q S= X X (-l) k ( q ) u(m+p+q-k;p,q) + (-l) q 2 q 2 (-1 ) r u(m+p-r; p, q) k=0 j=0 r=0 + (_ 1 } m-l 2q-l q-1 k m+p = 2 2 (-l) k ( q ) u(m+p+q-k; p,q) + (-l) q 2 q 2 (-1 ) r u(m+p-r; p, q) k=0 j=0 r=0 m + p + ( 1 ) m - l 2q-l + ( _ 1 } q - l 2 q (-1 ) r u(m+p-r; p, q) r=m-q+l q-1 k m+p = 1 1 (-l) k ( q ) u(m+p+q-k; p,.q) + (-l) P + q 2 q 2 (-l) " 1^!; p, q) k=0 j=0 p+q-1 + (-l) " 1 z ^ 1 + (.D +p+q-i 2q s (.i) 1 u ( i ; p< q )

7 1964 A GENERALIZATION OF FIBONACCI NUMBERS 283 Solvig for S, ad otig P+q-i V / i\i / \ / 0 P +C l eve, ) 2 (-1) u(i;p.q) = [ l ^ Qdd J = we get the result (3. 2). F r o m (3. 1) ad (3. 2) we ca obtai expressios yieldig 2 u(2i; p, q) ad 2 u(2i + 1; p, q). I the simpler case where q = 1, we fid 2~p-fl 2 (3.3) 2 u(2i + 1; p, 1) = I (u(2+p+2; p, 1) u(2i+t ;p,l) ad 2-p-T 2 (3.4) 2 u ( 2 i ; p, l ) = i - [ u (2+p+2; p, 1) -1-2 u(2i+t]; p, 1)] where 7] = 0 whe p is eve ad rj = 1 whe p is odd. it is simpler to start with I this case u(2i+l; p, 1) = u(2i; p, 1) + u(2i-p; p, 1), 2i > p = u(2i; p, 1), 0 <L 2i < p ad sum. We obtai i this way 2-p-t 2 (3.5) 2 u(2i+l; p, 1) = 2 u ( 2 i ; p, 1 ) + 2 u(2i + T); p, 1). Sice we also ca write

8 284 A GENERALIZATION OF FIBONACCI NUMBERS December 2+l 2 u(i; p, 1) (3.6) 2 u ( 2 i + l ; p, 1 ) + 2 u(2i; p, 1) = u(2+p+2; p, 1) -1 by (3. 1), the r e s u l t s (3. 3) ad (3. 4) follow by additio ad subtractio ad solvig for the sum. For p = 1 these results reduce to the well-kow relatios of Fiboacci umbers: (3. 2') X ( " ^ " " ^ i ^ - l + { - 1 > ~ 1 (3.3') 2 f 7. = U., - 1 s ' 2i 2+l (3.4') 2 f 9. 1 = f 9 x ' 2i-l 2 Theorem Let q = 1 ad defie u(i; p, 1) = 0 for i a egative iteger. The (3.7) u(+m; p, l) = u(; p, l)u(m; p, 1) + 2 u ( - l - i ; p, l)u(m-p+i; p, 1), p-1 where, m a r e aypositive itegers or z e r o. To prove this we ote first that this is true for ay positive iteger or zero ad m = 0. For ay positive iteger or zero ad 0 < m = k ^ p we have

9 1964 A GENERALIZATION OF FIBONACCI NUMBERS 285 P-l u(; p, l)u(k; p, 1) + X u ( - l - i ; p, l)u(k-p+i; p, 1) P-l = u(; p, 1) + X u ( - l - i ; p, 1) i=p-k +k-p-1 = u(; p, 1) + X u(j; p, 1) j=-p +k-p-1 -p-1 = u(; p, 1) + X u(j; p, 1) - X u(j; p, 1) j=0 j=0 = u(; p, 1) + u(+k; p, 1) - u(; p, 1) = u(+k; p, 1) where the sums have bee evaluated usig (3. 1). Hece (3.7) is true for ay positive iteger or zero ad m = 0, 1,..., p. For m = p+1 we get p - l u(; p, l)u(p+l; p, 1) + X u ( - l - i ; p, l)u(p+l-p+i; p, 1) p - l = 2 u(; p, 1) + X u ( - l - i ; p, 1) = u(; p, 1) + X u(j; p, 1) j=-p = u(+p+l; p, 1). Assume ow, fially, that (3.7) is true for ay positive iteger or z e r o ad m = 0,. 1,..., p,..., k where k >L p+1. The

10 286 A GENERALIZATION OF FIBONACCI NUMBERS December p-1 u(+k-p; p, 1) = u(; p, 1 )u(k-p; p, 1) + 2 u ( - l - i ; p, l)u(k-2p+i; p, 1) u(+k; p, 1) = u(; p 9 l)u(k; p, 1) + 2 u ( - l - i ; p, l)u(k-p+i; p, 1) Hece u(+k+l; p, 1) = u(+k; p, 1) + u(+k-p; p, 1) = u(; p, 1) [u(k; p, 1) + u(k-p; p, 1)] p u ( - l - i ; p, 1) [u(k-p+i; p, 1) + u(k-2p;p, 1)], p-1 = u(;p, l)u(k+l;p, 1) + 2 u ( - l - i ; p, 1) u(k+l-p+i;p, 1) But this is (3. 7) with m = k+1 ad the theorem is proved. For m =, equatio (3. 7) becomes (3.8) u(2;p, l) = u 2 (;p, 1 )+ 2 u (-2iL;p, l)+2 2 u(-i;p, 1 )u(-(p+l )+i;p, 1), 2 p odd ad (3. 9) u(2;p, 1) = u 2 (;p, 1) u(-i;p, 1 )u(-(p+l )+i;p, 1), P 7 F o r m = +1, equatio (3.7) becomes p - i 2 2 p eve ) u(2+l;p, 1) = u (;p, 1) u(-i;p, l)u(-p+i;p, 1), p odd ad

11 1964 A GENERALIZATION OF FIBONACCI NUMBERS 287 (3. 11) u(2+l;p, 1) = u 2 (;p, 1) + u 2 (- ; p, 1) -> u(-i;ps l)u(-p+i;p, 1), p eve Whe p = 1 equatios (3. 7), (3.8) ad (3. 10) reduce to the kow relatioships (3.7') f _^T = f,. f,_ +f f +m m+1 m (3. B') f, ^ = f 2 + f 2 2+l +1 (3. 10') f 9, = f f f f DIVISIBILITY PROPERTIES Theorem Ay p + q cosecutive t e r m s a r e relatively p r i m e. The t e r m s u(0; p, q),..., u(p + q - 1; p, q) a r e all uity ad so relatively p r i m e. Ay p + q cosecutive t e r m s cotaiig oe of these will have greatest commo divisor 1. Assume (u(; p, q), u( + 1; p, q),..., u( + p + q - I; p, q)) = d, where > p + q - 1. The because of (2. 2) it follows d (u( - 1; p, q), u(; p, q),..,, u( + p + q - 2; p, q)). Successive applicatios will show d (u(p + q - 1; p 5 q), u(p + q; p, q),..., u(2p + 2q - 2; p, q)). This cotais u(p + q - 1; p, q) so that d = 1 ad the theorem follows. Theorem The least o-egative residues modulo ay positive iteger m of (u(; p, q)} a r e periodic with period P ot exceedig m ". There is o preperiod. Each period begis with p + q t e r m s all uity. There a r e m possible least o-egative residues modulo m for each u(; p, q) ad m possible a r r a g e m e t s of residues i p + q cosecutive t e r m s. Sice by (2. 2) the residue of u(; p, q)

12 288 A GENERALIZATION OF FIBONACCI NUMBERS December depeds upo the residues of the precedig p + q t e r m s, after m P q t e r m s at most the residues m u s t repeat with a period P. Suppose u( + p; p, q) is the first t e r m such that the residues repeat ad a s - sume > 0. The u( + P + j ; p, q) = u( + j ; p, q) (mod m), j = 0, 1,..., p + q. I view of the r e c u r s i o formula, this shows u( P; p, q) = u( - 1; p, q) (mod m), a cotradictio to the assumptio u( + P; p, q) is the first t e r m such that the residues repeat. Thus = 0 ad there is o preperiod. Hece each period begis with p + q t e r m s each uity. As a example, we have residues (mod 7) for u(; 2, 1) r l l l r l l r l Here P = 57. Theorem 4. 3 Ay p r i m e divides ifitely may u(; p, q). If the period of the residues (mod m) is P, the m divides each of u(p P k; p, q), u(p P k; p, q),..., u(p - p + P k; p, q), k = 0, 1, 2,... Sice the residues a r e periodic it is sufficiet, to establish the first part of the theorem, to showthat a y p r i m e divides oe u(; p, q). Let m be ay give p r i m e or multiple of ay give p r i m e. The with P the period, u(p; p, q) = u(p + 1; p, q) =... = u(p + p + q - 1; p, q) = 1 (mod m).

13 1964 A GENERALIZATION OF FIBONACCI NUMBERS 289 F r o m the r e c u r s i o formula, q u ( P - l ; P,q) = 2 (-1) 1 ( q {) u(p p + q - i; p, q) q E X ( - I ) ' (]) = 0 (mod m) Hece m u(p - I; p, q). Similarly for u(p - 2; p, q),..., u(p - p; p, q). I the previous example 3 we ote 7 u(56; 2, 1), ad 7 u(55; 2, 1). Of course, 7 also divides other t e r m s, as the table idicates. REFERENCES 1. Basi, S. L., Geeralized Fiboacci Sequeces ad Squared R e c - tagles, A m e r i c a Mathematical Mothly, Vol. 70, 1963, pp * 2. Dickiso, David, O Sums Ivolvig Biomial Coefficiets, A m e r i c a Mathematical Mothly, Vol. 57, 1950, pp a Horadam, A. F., A Geeralized Fiboacci Sequece, A m e r i c a Mathematical Mothly, Vol. 68, 1961, pp M i l e s, E. P., J r., G e e r a l i z e d F i b o a c c i N u m b e r s ad A s s o - c i a t e d M a t r i c e s, A m e r i c a M a t h e m a t i c a l M o t h l y, Vol. 67, I 9 6 0, p p » R a a b, J o s e p h A., A G e e r a l i z a t i o of t h e C o e c t i o B e t w e e t h e F i b o a c c i S e q u e c e a d P a s c a l ' s T r i a g l e, T h e F i b o a c c i Q u a r - t e r l y, Vol. 1, O c t., , p p Netto: Lehrbuch der Kombiatorik, Teuber, Leipzig, 1901, p Hochster, Melvi, Fiboacci-type s e r i e s ad P a s c a l ' s triagle, P a r t i c l e, Vol. IV, 1962, pp Feiberg, Mark, New Slats, The Fiboacci Quarterly, Vol. 2, 1964, pp xxxxxxxxxxxxxxx