A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia 26506 Almost eveyone who woks with Fibonacci numbes knows that diagonal sums in the Pascal tiangle give ise to the fomula [?±] (D F. ( " - * " ' ), a>1. but not many ealize that (2) F 2n = f-/i*("-*-')'- a, o that \i\ (3) F 3n = 2 { " ' I ' 1 )4"- 1-2k, that these ae special cases of a vey geneal fomula given in 1878 by Edouad Lucas [5, Eqs. 74-76], [6, pp. 33-34]. As fa as I can detemine, fomula (2) fist appeaed in ou Fibonacci Quately as a poblem posed by Luline Squie [10] when she was studying numbe theoy at West Viginia Univesity. M. N. S. Swamy's solution invoked the use of Chebyshev polynomials. I was eminded of the fomula ecently when Leon Benstein [1] found the fomula again asked me about it. He used a new technique involving algebaic numbe fields. Fomulas (2) (3) genealize in a cuious manne. On the one h we have fo even positive integes F [T] (4) f 1 = H (~V k { n ~ k k' 1 )L"' 1 ' 2k. 2\. but on the othe h fo odd positive integes we get the same tems but with all positive signs (5), IT] F f = J2 ( n ~ k k- 1 )L n - 1-2k, k=1) ' 2\, whee L is the usual Lucas numbe defined by L n+1 = L n + L n -f, with LQ = 2, Lj = 7, this of couse in contast with F n +i = F n + F n --j FQ = O f F-J = 1. Fomulas (4) (5) may be witten as a single fomula in a cleve way as noted by Hoggatt Lind [4] who would wite n-u (6) T =11 (-V k( ' 1 > {"- k k- 1 )L" - 1-2k, 25
26 A FIBONACCI FORMULA OF LUCAS AND [FEB. valid now fo any positive integes nj > 1. Fomula (6) of Hoggatt Lind was posed as a poblem by James E. Desmond [11] solved by him using a esult of Joseph A. Raab [7]. The pecise same poblem was posed again by David Englund [12] Douglas Lind pointed out that it was just the same fomula. Fomulas (4) (5) wee obtained by Hoggatt Lind [4] by calculations using compositions geneating functions. Although they cite Lucas [5] fo a numbe of items they wee evidently unawae that the fomulas appea in Lucas in a fa moe geneal fom. Since L = F2 /F, fomulas (4)-(5) can be witten entiely in tems of f ' s. Lucas intoduced the geneal functions U, V defined by (7) U n = a =-f> V n = a n +b n, a b whee a b ae the oots of the quadatic equation (8) x 2 -Px + Q= 0, so that a + b = P ab = Q. When we have x 2 - x - 7 - ft we gets /? as (1 + ^Js)/2 (1 -yjs)/2 then U n = F n, V n = L n. One of the geneal fomulas Lucas gave is [6, pp. 33-34, note mispint in fomula] l"f] (9) ^ = (~V k ( n ~ k k- 1 ) Vf- 1-2 *!]*, which unifies (4) (5) is moe geneal than (6). Cuiously, as we have intimated, Hoggatt Lind do not cite this geneal fomula. Now of couse, thee ae many othe such fomulas in Lucas' wok. Two special cases should be paaded hee fo compaison. These ae (10) L n = (-1) k --V-- ( n - k ) L n ~ 2k f o m e, f] mi L. - 2 - ("-*) c - 2k «' - These can be united in the same manne as (4)-(5) in (6). Thus [I] (11.1) L n = (-7) k( ~ 1) -JLj {"- k ) L n ~ 2k. Thee is nothing eally mysteious about why such fomulas exist Thee ae pefectly good fomulas fo the sums of powes of oots of algebaic equations tacing back to Lagange ealie. The two types of fomulas we ae discussing aise because of [ ] (12) (- 1) k (" ~ *) (xy) k (x + Y 2k = X -^X-, fomula (1.60) in [3], X V
1977] ITS SUBSEQUENT MANIFESTATIONS AND REDISCOVERIES 27 f - 1 L2J (13) ^ ( - j ) * " { n - k )(xy) k (x+y 2k = x n +y n, n * fomula (1.64) in [ 3], familia fomulas that say the same thing Lucas was saying. The eason it is not mysteious that (2) holds tue, e.g., is that Z^/? satisfies the second-ode ecuence elation F 2n+2 = 3F 2n ~ F2n-2 with which we associate the chaacteistic quadatic equation x 2 = 3x - / so that a fomula like (2) must be tue. Fo fomula (4) with = 4 we note that F4 n+ 4 = 7F4 n - F4 n _4. In geneal in fact, (14) F n+ = L F n -F m. fo even, o F m+ + F m - = L F m, (15) F m+ = L F n + F n - ioodd, o F m+ - F m. = L F m. Regulaly spaced tems in a ecuent sequence of ode two themselves satisfy such a ecuence. Set u n = F n to see this fo then we have (16) u n +i = L u n ±u n -i, with z 2 = L z± 7, so we expect a pioi that u n must satisfy a fomula athe like (1). Fomulas like (12) (13) give the sums of powes of the oots of the chaacteistic equation, whence the geneal fomulas. Fomula (12) coesponds to (B.1) (13) coesponds to (A.1) in Daim's pape [2] which the eade may also consult. Anothe inteesting fact is that these fomulas ae elated to the Fibonacci polynomials intoduced in a poblem [9] discussed at length by Hoggatt othes in late issues of the Quately. These ae defined by f n (x) = xf n -f(x) + f n - 2 (x), n > 2, with fi(x)= 1 f2(x) = x. In geneal m (17) f n (x)= {"- k k- 1 )x"- 2k - 1. whence fo odd we have by (5) that (18) f {L ) = F -P. ' Many othe such elations can be deduced. Finally we want to note two sets of invese pais given by Riodan [8] which he classifies as Chebyshev invese pais: i] (19) «W = ( - 1>k ^k ( n k k )9( n ~ 2k ) if only if [I] (20) g(n) = (" k )f(n-2k);
28 A FIBONACCI FORMULA OF LUCAS AND [FEB. tfi (211 IM- X. TlfH» ' " - 2kl if only if [i] (22) g(n) = (-if ( n ~ K ) f(n-2k). Applying (19) (20) to (10) we get the paticulaly nice fomula (23) L n = (" k ) L (n-2k), even. Using (21) (22) on (4) we get the slightly moe complicated fomula u i] (24), ^ i t n ^ f o ^ a n-k+1 \kl F k^o I do not ecall seeing (23) o (24) in any accessible location in ou Quately. If we let /--> 0 in (4) we can obtain the fomula (1.72) in [3] of Lucas, which is also pat of Desmond's poblem [11] who does not cite Lucas, I'l (25) n= (~1) k ( n - k k- 1 )2"- 2k - 1, n>1. It is abundantly clea that the techniques we have discussed apply to many of the genealized sequences that have been intoduced, e.g., Hoadam's genealized Fibonacci sequence, but we shall not take the space to develop the obvious fomulas. It is hoped that we have shed a little moe light on a set of athe inteesting fomulas all due to Lucas. REFERENCES 1. L. Benstein, "Units in Algebaic Numbe Fields Combinatoial Identities," Invited pape, Special Session on Combinatoial Identities, Ame. Math. Soc. Meeting, Aug. 1976, Toonto, Notices of Ame. Math. Soc, 23 (1976). p. A-408, Abstact No. 737-05-6. 2. N. A. Daim, "Sums of n f Powes of Roots of a Given Quadatic Equation," The Fibonacci Quately, Vol. 4, No. 2 (Apil, 1966), pp. 170-178. 3. H. W. Gould, "Combinatoial Identities," Revised Edition, Published by the autho, Mogantown, W. Va., 1972. 4. V. E. Hoggatt, J., D. A. Lind, '"Compositions Fibonacci Numbes," The Fibonacci Quately, Vol. 7, No. 3 (Oct., 1969), pp. 253-266. 5. E. Lucas,"The'oiedes FonctionsNume'iquesSimplementPe'iodiques/'/l/77e. J. Math., 1 (1878), pp. 184 240;289-321. 6. E. Lucas, "The Theoy of Simply Peiodic Numeical Functions," The Fibonacci Association, 1969. Tanslated by Sidney Kavitz Edited by Douglas Lind. 7. J. A. Raab, "A Genealization of the Connection between the Fibonacci Sequence Pascal's Tiangle," The Fibonacci Quately, Vol. 1, No. 3 (Oct. 1963), pp. 21-31. 8. J. Riodan, Combinatoial Identities, John Wiley Sons, New Yok, 1968.
1977] AND ITS SUBSEQUENT MANIFESTATIONS AND REDISCOVERIES 29 9. Poblem B-74, Posed by M. N. S. Swamy, The Fibonacci Quately, Vol. 3, No. 3 (Oct., 1965), p. 236; Solved by D. Zeitlin, ibid.. Vol. 4, No. 1 (Feb. 1966), pp. 94-96. 10. Poblem H-83, Posed by Ms. W. Squie, The Fibonacci Quately, Vol. 4, No. 1 (Feb., 1966), p. 57; Solved by M. N. S. Swamy, ibid., Vol. 6, No. 1 (Feb., 1968), pp. 54-55. 11. Poblem H-135, Posed by J. E. Desmond, The Fibonacci Quately, Vol. 6, No. 2 (Apil, 1968), pp. 143-144; Solved by the Popose,/M/ Vol. 7, No. 5 (Dec. 1969), pp. 518-519. 12. Poblem H-172, Posed by David Engl, The Fibonacci Quately, Vol. 8, No. 4 (Dec, 1970), p. 383; Solved by Douglas Lind, ibid., Vol. 9, No. 5 (Dec, 1971), p. 519. 13. Poblem B-285, Posed by Bay Wolk, The Fibonacci Quately, Vol, 12, No. 2 (Apil 1974), p. 221; Solved by C. B. A. Peck, ibid., Vol. 13, No. 2 (Apil 1975), p. 192. ******* [Continued fom page 24.] (iii) ( ^ - 7 ) G 2 + G 2 H = G 2n +1 (n > 1) (iv) G 2 n+2- (P-flf'G 2 = G 2n+2 (n > 1) =0 (vi) { P -J- L )T, = G n+2-1 (n > V. =1 The poofs of the above esults, which ely essentially on equations (2), (3) (5), togethe with a - j 3 = V p, a+p.= 1 a(s =-(&- ), ae faily staightfowad left to the eade. Of couse, esults such as these ae not new. Fo example, (ii) was poved in a slightly moe geneal fom by E. Lucas as ealy as 1876 (see [1] page 396). Finally, tuning to the vetical sequences in the table given ealie, it follows fom (v) that the sequence unde G n (n > 1) is given by is'c-;- )'*-"'} (6) \ X [ n 'l- )(k-u \ <k> 1), t 1 =0 =0 J so that fo example the sequences unde G4 G5 ae {2k - 7} {k + k- / }, espectively. Altenatively, instead of using (6), we can apply the Binomial Theoem to (2) obtain the geneal vetical sequence in the fom ~~n~l 2 { n )(4k-3) ( - 1J/2 } (k > 1).. 2 =1 J odd REFERENCE 1. L. E. Dickson, Histoy of the Theoy of Numbes, Vol. 1, Canegie Institution (Washington 1919).