DIRECT CALCULATION OF k-generalized FIBONACCI NUiBERS

Transcription

1 DIRECT CALCULATION OF -GENERALIZED FIBONACCI NUiBERS IVAN FLORES Broolyn, New Yor SUMMARY A formula i s developed for direct calculation of any -generakzed F i b - onacci number u., s K without iteration. DEFINITIONS The ordinary Fibonacci number u. * is defined by () V ; u. = u. + u. (i > ) * ]-l, J- U ~ 7 with the additional conditions usually imposed () u. = ; u U =. The -generalized Fibonacci number u., i s defined a s the sum of i t s p r e d e c e s s o r s ( ' u j, U j~i s U j- # U j- s i=j- together with the initial conditions () u j j = ( L j L - ); V l j = A table of u.,» K from = to 7 and j = to is found in Table. 9

2 6 DIRECT CALCULATION OF -GENERALIZED [Oct i 6 7 P Table Fibonacci Numbers u., for Various Values of i and ] TERM RATIO The ey to direct calculation is the existence of a fixed ratio r, between successive u.,! s so that in the limit we have () lim ]+i ; n >oo u., If such a ratio can be founds an approximate calculation is simple. Vorob'ev [s] has shown that for =, this requires the solution of (6) q^ = q^" + q^~ which for q f reduces to (7) q - q - = for which the roots are.() r i = Lz^l * -.6 and r = ^ y ^ - *.6, where s& means "approximately equal to* " If f is any Fibonacci sequence obeying the difference equation f f n " f n-i = t h e n f n h a s t h e f o r m < s e e E ]* (9) f n = b i r i + b r

3 967] FIBONACCI NUMBERS 6 Since ( r ^ <.7, jr^j < \ 9 so that r p < l / n e Hence there exists an such that for all n > N s u n is the greatest integer to b rf, and we write N () u js * V > N) To evaluate the constants h t and b, we use the initial conditions (ID b l + b = u j = h T + b r = u l s =, which yield K - () U bi = V :, b = V An exact expression for u. is hence the familiar Binet form () u = hl V H-(^)] GENERALIZATION To find an expression for the -generalized Fibonacci n u m b e r s, let us first see solutions to () which form a geometric p r o g r e s s i o n say aq. Then () leads to a general form of (6), () Thus () J J- j ~, j - aq J = aq J + aq J + + aq J j -, -i aq (q - q q - ) =. Since we a r e looing for solutions which a r e not identically zero,, a s s u m e a ^ and q f o Therefore we see we can (6) -i q - q - q - =

4 6 DIRECT CALCULATION OF -GENERALIZED [Oct.,th This degree equation has complex roots, say r t ^, r ^, r,,. Now Miles [j has shown that these roots are distinct, that all but one of them lie within the unit circle in the complex plane, and that the remaining root is real and lies between and. Hence with a suitable choice of subscripts we may write (7) Kl-< x (^ ^ - x ) > () * r, * Since the roots are distinct, the Vandermond determinant (9) r i, i, r, r, -i r l, -i r, f, r.,. rf,. -i r, and Jese fj has shown that the general solution can be written () u. *-* I I, i = i j, E b - r - To evaluate the constants b., we use the initial conditions i () % Vii = (m = ' ',, ' - ) > = -i, i=l? V?-- = This system has a unique solution by (9) which can be found using Cramer's rule. This yields

5 967] FIBONACCI NUMBERS < ) b i - n < r i, ' V r l QF= so that () becomes () v = M (r i^ - W W i Recalling (7) and (), we remar that as j becomes large ri, becomes the dominant term in (), so that as before there exists an N such that for all j > N, u.. is the nearest integer to b, rj",. We may therefore write () u j ; * b (j > N). It follows from () that u. () lim - C M = r. u.,, J-> J, and more generally (6) lim! & = r m APPROXIMATIONS We first note that as >oo the sequence u.,, approaches the geo- J-K,K metric progression of powers of two,,,,, 6,, 6, as can be seen from Table. It follows that

6 6 DIRECT CALCULATION OF -GENERALIZED [Oct. (7) lim r =. ^oo ' See Table for calculated values of the principal root r., for = to 9, which gives striing verification of (7). Table Fibc onacci Roots r F r o m () we get then () lim lim - > o o j-9>oo u, *' K =, U j j which was stated in an equivalent form by P. F. Byrd [ l l. We shall now show - that b, is approximately r,, in the sense that

7 967 (9) FIBONACCI NUMBERS lim b. / r ". =., ^ /, 6 To prove (9), first recall that ( r, ~ ^ ^ " ( r, - r - i, ) Since -i x - x x - = (x - r l j ) (x - r ),' ' and f(x) = ( x - l ) ( x - x " x - ) = x + - x + l = ( x - l ) ( x - r i ) -. - ( x - r.. ),,' we find., f ' ( r ) ) = ( + l ) r ^ - r = ( r ^ - ihr^ - r > ) -. ( r ^ - r _ > ) Hence, " ( + i ) r ) - r ; from which (9) follows, since r, + - r fe = -. Then for sufficiently large j and we may write () u j j * r. Call the approximation for u.. in () u!,. Then using (), the e r r o r committed by this approximation is () w., =.. - ul, j, j, j s - i i = i By (7) Ir., < for L i -, so the triangle inequality shows

8 66 () DIRECT CALCULATION OF -GENERALIZED FIBONACCI NUMBERS -i. -i w., ^ E M l r - i I < E b- j ~ j I i i,.tj I i Oct. Note that the first inequality in () shows that () lim w.. =, J->oo J so that for fixed the e r r o r tends to zero as j becomes large, giving formal justification to (). EXTENSION In the near future tables of b, will be p r e p a r e d by computers. These, together with r,, will provide an excellent approximation for u., using an analytic procedure. The author e x p r e s s e s his appreciation to D a A. Lind and V Eo Hoggatt, J r., for helping in p r e p a r i n g this paper. REFERENCES. P. F. Byrd, Problem H-6, Fibonacci Quarterly, (96),. Solution, (967).. Mar Feinberg, "New Slants, " Fibonacci Quarterly, (96), -7.. Mar Feinberg, "Fibonacci-Tribonacci, n Fibonacci Quarterly, (96), No., 7-7,. J a m e s A. Jese, "Linear R e c u r r e n c e Relations, P a r t I, " Fibonacci Quarterly, (96), No., E. P. Miles, J r., "Generalized Fibonacci Numbers and Associated M a t r i c e s, " American Math. Monthly, 67(96), N. N. Vorob'ev, Fibonacci Numbers, Blaisdell, New Yor, 96. * * *