ANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION

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1 ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals have give idicatio of a reewed iterest i the well-kow Fiboacci sequece, amely (1) 1, 1, 2, 3, 5, 8, -, C, where C C ' + C. > 3, with C 1 Cl -i -2 * 2 1. Some recet geeralizatios have produced a variety of ew ad exteded results. A search of the literature seems to reveal that efforts to geeralize the Fiboacci sequece have cosisted of either (a) chagig the recurrece relatio while preservig the iitial terms, or (b) alterig the iitial terms but maitaiig the recurrece relatio. will be employed here. A combiatio of these two techiques Heretofore, all geeralizatios of the Fiboacci sequece appear to have restricted ay give term to beig a fuctio (usually sum) of the two precedig terms. I this paper we shall exted this by cosiderig sequeces i which ay give term is the sum of the three precedig it. Sice the set of all algebraic itegers, i.e., all y such that y satisfies some moic polyomial equatio, / \, c\ p(x) x + a x + *» < > + a i X + aq o, with itegral coefficiets ad of degree greater tha zero, is a itegral domai uder the operatios of additio ad multiplicatio, it was cosidered worthwhile to examie sequeces i which the iitial terms (hece all succeedig terms) are algebraic itegers. It will be show that certai special cases of such sequeces are especially useful i the examiatio of the more geeral case. 29

2 21 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. 2. THE GENERALIZED SEQUENCE { } Specifically we cosider the sequece (2), i, 2,,, where, i, 2 a r e give, a r b i t r a r y algebraic itegers, ot all zero, ad (3) + +. > 3. - l -2-3' It will also be coveiet to cosider a compaio sequece, so to speak, (4) { R j R, Ri, R 2.. R > ", where R i -, Ri 2 - i. ad for > 2, (5) R, + - l -2 F r o m (5) ad (3), whe > 5, we have R + ( + J + ( + ) + ( + J - l R + R + R - l -2-3 Usig (5) ad (3) further, we have R4 R3 R3 + R 2 + Ri R 2 + Rj + RQ Hece for > 3, (6) R R + R + R. -l -2-3

3 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE 211 Thus R is actually the special c a s e of (2) i which R i -, -Rl p 2 - p i» R 2 p + l p T e o evidet i the developmet of j usefuless of the sequece j R i will be I that follows. Two other special c a s e s of (2) should be metioed at this time; amely the c a s e s i which, j 2 1 ad 1, ~ t, 2 1 r e - spectively, to give the sequeces (7), 1, 1, 2, 4, 7, 13, 24, 44,, K,, ad (8) 1,, 1, 2, 3, 6, 11, 2, 37, e ', L, '. We s e e immediately that L Ki - K, Lj K 2 - K^, ad for > 2, (9) L K + K. -l -2 Hece we might call J K } a -type sequece ad J L a R - t y p e sequece,, Til. The sequece JK was defied ad discussed briefly by M. Agroomoff The followig t h r e e relatios ivolvig various t e r m s of this sequece w e r e discovered ad proved by him: (1) K K K + (K + K )K + K K, x ' +p p+l p - l p - i p -2 (11) x ' K 2ri <* K 2 - i, + K (K + K + K ), x +i - 1 _ 2 ' (12) K, K 2 + K 2 + 2K K x ' 2-l - l - i -2 T h e r e is oly oe basic idetity h e r e because the latter two a r e evidetly special c a s e s of the first oe upo settig p ad p - 1 respectively. F u r t h e r, it was cojectured i [ l ] that eve though the sequece (7) was a Fiboacci-type sequece, it quite possibly would p o s s e s s few of the i t e r e s t - ig p r o p e r t i e s which the Fiboacci sequece has, ad eve if it should, such properties would be much m o r e difficult to fid due to the m o r e complex ature of the r e c u r r e c e relatio determiig the sequece.

4 212 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. We tur ow to a ivestigatio of the sequece (2) ad cosider, amog other facts, how (1), (11), ad (12) occur as special c a s e s of m o r e geeral relatios,, arallelig the usual treatmet of the Fiboacci sequece, we obtai a closed expressio for sice J I satisfies a differece equatio. Thus (13) B ^ + B 2 x* + B 3 x^ where xj,, x 2, x 3 a r e the t h r e e distict roots of the equatio x 3 - x 2 - x - 1, ad Bj, B 2, B 3 a r e costats depedig o these roots as well as, i, 2, ad a r e determied by the system i Bt + B 2 + B3 BtX! + B 2 x 2 + B 3 x 3 i 2 Bpdj + B 2 x + B 3 x 3 2. The values of x*, x 2, x 3, Bj, B 2, B 3 a r e such as to make (12) too c u m b e r - some to be of ay further practical u s e i the succeedig developmet ad hece will ot be writte h e r e. A much m o r e useful way of represetig the r e c u r r e c e relatiofor j J may be foud as follows: I the otatio of vectors ad m a t r i c e s, we have by (3), ("s" p 2 L i. " 1 1 1"] 1. 1 oj 2 1 i LOJ pr p 3 " l i i i 1 ps" p 2 " i l l " 1 2 "p 2 ] i L 2. _ 1 Oj L I 1 _.oj

5 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE ad by fiite iductio (14) rp -i " " -2 " 2 ] l. p o J 213 F u r t h e r, a simple iductio proof gives (15) "1 1 f 1 p 1 q. K, _, L.. K +1 +l K L K 1 -l - K -i L -i K -J so it might be said that j K I ad i L a r i s e " aturally" i t h e ivestigatio of { j. Usig (14) ad (15), we fid for, p positive itegers that (16) " +p "+p-i L +p-2. L K "K p+l ^ p+i ^ p K L K.,, x H - 1 II - i K L K p-i p-i p - 2 J U -2 from which we immediately see that (17) (18) ^ K ^J + L, + K +p p+l p+l - l p -2 K + L + K 2 +i +l -l -2 K + (K + K ) + K o +l -l - l -2 (19) K + (K + K J + K. 2-i -l - 2 ' - l - i -2 Now settig, j 2 1, we have (1), (11), (12) as special c a s e s of (17), (18), ad (19), respectively, Sice

6 214 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. +p +p-i K... L., J K. p+r+1 p+r+1 p+r K ^ L ^ K _,_ p+r p+r p + r - i - r " - r - i we also have L +p-2 J p+r--i p + r - i p+r-2j - r - 2 (2) u. K ^ _, + L ^ + K ^ +p p+r+i - r p+r+1 - r - i p+r - r - 2 for, p, r positive itegers, r < - 2, Similarly for, h, k positive itegers, we ca show that (21) +h+k * h+k+i + L h +k+l - i + +k - l Usig (2) ad (21), we have the followig useful expressio: (22) "+h+k L +h K -,-2 xx u h+k+i hi+k+i^+k 1 ri I r " p 2 K h+i L h+i K h 1 J 1 i L i _. p o It ca be show quite easily that the sequece (23) l f R 2, 2, R 3, 3, ',, R + 1, is geerated by the matrix ri i [ i i 1 1 OJ that is, (24) r p i R. L ri-i-j "I 1 <r -2 " 2 l R2. i J

7 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE 215 It is a iterestig ad useful fact that this matrix is the traspose of the geeratig matrix for J }. Usig (24) i away aalogous to that i which we established (21), we prove that (25) a.^, K. M _^ + K, _,. R + IC, A +h+k h+k+l h+k li+k-i -l (26) R +h+k L h+k+i + L h+k R + ^ k ^ - i two relatios which are ot oly iterestig i themselves but which also give +h+k K h+k+i K h+k K h + k - i ] ri i o" 1 1 <U " p 2 (27) R +h L h+i \ L h - i 1 1 R 2 «- 1 J [i o o,. i i the form hi order to defie for egative, we use (14) for > writte (28) r p - k + i L +2 J " 1 " "ol 1 _1 1 1_ i J>i\ Replacig by - i (28), we have for >, (29) r p i L -+2- i ro i o* 1 I I \ i i "o" i fol 1 i. J which together with (14) determies for all sice? j, 2 are give. The same result is obtaied upo replacig by - i (3) to get (3) J J J. - J -+1. ^ > 8

8 216 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. R is also defied for egative by (29) ad (3) sice R + -l -2 This allows us to remove the restrictio placed o, p, r, h, k above. 3 LINEAR SUMS A large umber of what we shall call liear sum relatios o t e r m s of the sequeces j R ad j f w e r e foud ad proved. Sice a exhaustive list is ot our aim, oly a few of the m o r e iterestig oes a r e listed. No proofs will be give h e r e sice the proofs may all be made r a t h e r easily by fiite iductio. 1 (31) T. -i ( ) u f-' l 2 +2 * 1 3-i (32) E 3i E?i + o, ii io < 33 > E ^3i s " o» ii (34) E R 3i+l s+l " i ii T h e s e relatios obviously have special cases for the sequeces K ad L F o r example (33) becomes (33 E Lsi K 3. ii

9 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE QUADRATIC AND CUBIC RELATIONS A attempt to parallel the quadratic relatios of the Fiboacci sequece failed. A differet approach was e c e s s a r y ad this was foud i the u s e of the v e c t o r - m a t r i x represetatio of. We have the followig iterestig quadratic form: (35) ^ + ^ i _ _ 2 + R i 2 _ 4. The proof of (35) follows by cosiderig the left side of the relatio as the s c a l a r product of the vectors f p, R,,1 ad f p,, 1 ^ L - i j L J - i ' -2 J (recall R + ), ad the usig (14) ad (24), we have - i -2 p2 + p2 + 2 f p, R, J - i - i -2 I - l j ' - i -2-1 ri i r T 2, R 2, J "2 "j i L i. oj [p 2, R 2J t] rp 2 - p L p 2- " 2 1 -*J 2?2-2 + Rj^Z-S + l 2-4 F o r, A 2 1, (35) becomes (35* K 2 + K 2 + 2K K K H - l -l -2 2-l which is (12), It was show that (12) is also a special c a s e of (19), but (35) is ot obtaiable from (19) or vice v e r s a. Oe of the most iterestig relatios ivolvig t e r m s of the Fiboacci sequece is the oe C c - C 2 ( - l ) - i +i

10 218 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. T h e r e is a relatio of this ature for the sequece j 1; however as may have bee suspected, it has a cubic r a t h e r tha a quadratic form,, The desired relatio is (36) ^ -3 - l -2 +i H - 2 +i - l -3 - l f i i !i - 2 i 2 - ^ Before provig (36), we ote that for, V 1 2 1, (36) becomes (37) K 2 K + K 3 + K 2 K - K K K - 2 K K K i -2 +i +l - l -3 - l -2 sice The proof of (37) follows from (9) ad (15) by the use of determiats K 2 K + K 3 + K 2 K _, - K ^ K K - 2K K K -3 - l -2 +l +i - l -3 -l -2 K K K +l - l K -2 - l K K K - l -3-2 K L ^ a., K +l +l K L K H - l K L K - l - l roof of (36): Eve though (36) may be verified i v e r y much the s a m e maer as (37), we adopt a differet method of proof sice this is m o r e easily used i a geeralized versio of (36). F i r s t, we state the followig l e m m a whose proof the r e a d e r ca readily supply, Lemma: Let A be ay 3 x 3 m a t r i x ad let x ad y be three-dimesioal vectors; the the c r o s s product (Ax) X (Ay) is equal to the cofactor matrix of A multiplied by x X y; i» e., (Ax) X (Ay) - (cofactor A) (x X y). Now the left side of (36) ca be cosidered as the triple s c a l a r product of the t h r e e vectors,., 1,., 1, ad fp,, 1. L +l - i j [_ - l -2-3j _ " 1 " 2 J By (14) ad the lemma,

11 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE 219 rp i - i -2 L - 3 J X " L - i * 1 1 f -3 " 2 " 1 i o - X " i l l " " J v\ i Therefore p 2 p -3 "-i + : -2 +l r p -1 +i L -i-, - 2 +i - l -3 -i -2 " - i " -2 L -3 J [ p 4, p s> p 2 ] X rp 1 - i. -J ri 1 o" L "? !_ p s p, 3-^ l 2 p i ( 2 i - 2 ) + 3( 3 o - 21) + 2 (! - 31). J which reduces to the right side of (36) becomes Example: Suppose we let, I > 1 1; the the right side of (36) o - 2o + 2. Settig this expressio equal to zero ad solvig for 2, we see that t h e r e exist algebraic itegers, say 3 l 3 2 s such that for the sequece J \, p 2 p + p 3 + p 2 p -3 - l -2 +l + 2 +l - i -3 - i -2

12 22 ANOTHER GENERALIZED FIBONACCI SEQUENCE [Oct. The l e m m a ad (22) may be used a s i the previous method of proof to show that for h, k,, m, t itegers (38) + + v ; +h +m +h+k+t +h+t +h+k+m +t +h+k +h+m h+m +h+k+t +m +h+k +h+t +h +t +h+k+m (K h L h+k+i K h + k L h + i ) [ t + 2 ^ l m ' o p m + i ) + t+i(o p m+2 2 m ) + p t ( p 2 p m + i - p i p m + 2 ) ] T h e r e a r e may iterestig special c a s e s of this relatio,, a few. If, t 2 1, (38) becomes We metio (39) K,,K. K.,,,,, + K K ^ J K ^UjLl ^ + K ^ K ^ ^ K ^, ^ v ' +h +m +h+k+t +h+t +h+k+m +t +h+k +h+m +h+m +h+k+t +m +h+k +h+t +h +t +h+k+m v h h+k~i h+k h - l / v t-i m t m - r If k h t, m 1, (39) becomes (4) K, K _,_, K., + K K K _, + K ^. K. ^ K _ - K K ^. K.. v ; +i +h +3h +2h +2h+i +h +h+i +2h +h+i +3h - K K 2 _ K 2, K _,_,_, K. K, K, - K 2 K, ; +i +2h +h +2h+i - i h 2h-l h-i 2h ad if t h, k m - h, (39) reduces to (41) K K _,_, K ^ + 2K ^, K ^ K ^, ^ - K K 2 ^ - K 2 K _ v ' +2h +2m +h +m +h+m +h+m +m +2h - K 2 K ^ - -(K.K - K K J 2 +h +2m \ h m - i m - i ' that I o r d e r that the above r e s u l t s be valid, we must choose h ad k so K K + k - i " K h + k K h - i " K h L h+k+i " K h+k L h+i * '

13 1967] ANOTHER GENERALIZED FIBONACCI SEQUENCE 221 for i the proof of (38), we a s s u m e that the matrix K h+k+i L h+k+i K h+k h+i L i L h+i K h is o-sigular Usig (27) we ca fid relatioships ivolvig t e r m s of both the sequeces R J ad j which reduce to a expressio idepedet of For e x - ample, it may be proved that \ ) +h+k+t^ -tfi +m +h+r +h+t^ +h+k+m +m +h+k' 4- p C R - R ] ~ ^ h + k - i ^ ' V k V i ' +t v +h+k +h+m +h +h+k+m [p^^m - m+i) + t + 1 ( m ^ - 2 m ) + t ( 2 m + i " p i p m + 2 ) ] It should be oted that o t e r m s of the sequece j R I appear o the right side of (42) ad also that the secod factor o the right side of the equality sig i (42) is the same as the secod factor o the right side of (38). 5. MISCELLANEOUS RESULTS We coclude with some miscellaeous r e s u l t s. The followig limitig relatios may be established usig (13) ad the fact that r l9 r 2, the two c o m - plex roots of x 3 - x 2 - x - 1, a r e such that r, ro < 1 (43) (44) V limit >oo ' limit - i > oo "+h V3V V V V 3 yg3" V Vg3 3 )

14 222 ANOTHER GENERALIZED FIBONACCI SEQUENCE Oct By iductio the followig theorem may be established: T h e o r e m : F o r every positive, K 4 K A _ t (mod 2) K 4-2 K ( m o d 2 ) K 4 (mod 4) If we let D(, j, 2,, ) be the determiat o i 2 i.., I ^ 2 +i +i +2 m it ca be show that for > 3, D(, l f 2, -, ) This material is take from Some Geeralizatios ad Extesios of the Fiboacci Sequece, a thesis submitted to the Uiversity of ittsburgh by the first author i partial fulfillmet of requiremets for the h D. degree. REFERENCE 1. M. Agroomoff, p M Ue s e r i e r e c u r r e t e, " Mathesis, ser. 4, vol. 4 (1914),