AN ALMOST LINEAR RECURRENCE. Donald E. Knuth Calif. Institute of Technology, Pasadena, Calif.

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1 AN ALMOST LINEAR RECURRENCE Doald E. Kuth Calif. Istitute of Techology, Pasadea, Calif. form A geeral liear recurrece with costat coefficiets has the U 0 = a l* U l = a 2 " ' " U r - l = a r ; u = b, u, + b~u b u, > r r -r The Fiboacci sequece is the simplest o-trivial case. however, the followig sequece: a) V 1 ; Cosider, ^ = *-l +^[/2J ' > 0 I this case, successive terms are formed from the previous oe byaddig the term "halfway back" i the sequece. This recurrece, which may be cosidered as a ew kid of geeralizatio of the Fiboacci sequece, has a umber of iterestig properties which we will examie here. The sequece begis 1/2, 4, 6, 10, 14, 20, 26, 36,.... It is easy to see that all terms except the first are eve, ad furthermore 0 2k-1 2k is divisible by 4 if ad oly if = 2 (mod 2 ) for some k > 1. We leave it to the reader to discover' further arithmetic properties of the sequece. The sequece <$> has a iterestig combiatorial iterpretatio: < is precisely the umber of partitios of the umber 2 ito powers of 2. For example, 6 = = = = = = , ad <t> = 6. To verify this iterpretatio, let P(m) be the umber of partitios of m The preparatio of this paper was supported, i part, by NSF grat GP-212. Ackowledgemet is also made to the Burroughs Corporatio for the use of a B5000 computer. 117

2 118 AN ALMOST LINEAR RECURRENCE April ito powers of 2. If 2 = a + a a, where a 1 _\ a >.. A a-i ad each a. is a power of 2, there are two cases: (i) a = 1; the 1 K. a a is a partitio of 2-l; (ii) a > 1; the a, /Z + a? / a, /2 is a partitio of. Coversely, all partitios of 2 are obtaied from partitios of 2~l ad i this way, so P(2) = P(2-1) + P(). We also fid P(2+1) = P(2) by a similar argumet; here oly case (i) ca arise sice is a odd umber. These recurrece relatios for P, together with P(l) = 1 ad P(2) = 2, establish the fact that <t> = P(2). The same sequece also arises i other ways; the author first oticed it i coectio with the solutio of the recurrece relatio (la) M(0) =0 M() = + mi 0 4 k < (2M(k)+M(-l-k)) for which it ca be show that M() - M(-l) = m if 0 < 2 < < >,, N x ' ' m - ^r+1, a d ^V^^-L^Z-ll Recurreces such as (la) occur i the study of dyamic programmig problems, ad they will be the subject of aother paper. Let us begi our aalysis of <t> by oticig some of its most elemetary properties. (2) * = 2 ( * «). By applyig the rule (1) repeatedly, we fid Aother immediate cosequece of (1) is ( x3 ) ' 4>l 2 ' <*> 2+l 9 M *, 2-1 T = < * 2. The sequece <t> grows fairly rapidly; for example, 4» A = *IOOOO = X 1 2 I fact, we ow show that <t> grows more rapidly tha ay power of :

3 1966 AN A L M O S T L I N E A R R E C U R R E N C E 119 T h e o r e m 1. F o r a y p o w e r k, t h e r e i s a i t e g e r N, s u c h t h a t k k 0 > for a l l > N,. k k+1 * 1 k+1 P r o o f : L e t N be s u c h t h a t (2 +1) > (2 + ^ ), ad let a = m m ( 0 / ). N N T h e by i d u c t i o 0 k+1 A a for a l l A N, s i c e t h i s i s t r u e for N 4 < 2N, ad if > 2N * = *-l + <^[/2] ^ t * - 1 ) * * 1 + j > / 2 ] k + 1 ) ^ // i \ k + 1.,11-Lk+l 1 k + l > 1 x k + l. 1 Nk+l > a ( ( - l ) + (-j-) ) = a(l + - E T r ) ( - l ) - a(l + ^ ) (-1).,_ ^ 1. k Nk+l k+1 > a(l + r ) (-1) = a - 1 If w e c h o o s e N, $> 1/a ad N, > N, the p r o o f i s c o m p l e t e. K. K. We ow c o s i d e r the g e e r a t i g fuctio for 0. L e t & (4) F(x) = x x 2 + <*> x +... N o t i c e t h a t ( l + x ) ( F ( x 2 ) = < + 0 Q x + 0 l X 2 + ^> 1 x x 4 + <*> 2 x = 0 Q + ( 0 r 0 o ) x + ( ) x 2 + ( ) x 3 + ( ) x 4 +. t h u s = ( l - x ) F ( x ) ; F(x) = ±E F(x 2 ) = ill2hl±4- F(x 4 ) X ' X ( l - x ) ( l - x * ) We h a v e t h e r e f o r e l*\ VM - ( l + x ) ( l + x 2 ) ( l + x 4 ) ( l + x 8 ) _ 1 \ D ) r v x / - 7 x ~ 7? 2 g (i-x)(i-x A )(i-x*)(i-x )--- (i-kr(i-x)(\-x)(i-x)-

4 120 AN ALMOST LINEAR RECURRENCE April F r o m this form of the geeratig fuctio, we see that F(x) coverges for x < 1. (As a fuctio of the complex variable z, F(z) has the uit circle as a atural boudary. ) It follows that lim sup \/<& = 1 > i. e. the sequece <t> grows m o r e slowly tha a for ay costat a > 1 0 This is i marked cotrast to liear r e c u r r e c e s such as the Fiboacci umbers 0 I the remaider of this paper we will determie the true rate of growth of the sequece that <t> ; it will be proved by elemetary methods 1 2 I <t> * -j -j- (I ), I 4 x ' 4 ( l ) 2 + o ( ( l ) 2 ) (6) <f> = e The techiques a r e similar to others which have bee used for d e t e r - miig the order of magitude of the partitio fuctio (see F2J ). We s t a r t by observig that ad hece by differetiatio ^ k I F(x) = -l(l-x) + V ^ ( - l ( l - x 2 )) k=0 E v E E r=l k=0 r=l? k 2 r x oo JS^ ^-E"'-, A 2 ' r=l k=0 r=l = 2 + 4x + 2x 2 + 8x 3 + 2x 4 + 4x $, x k ~ l +. k

5 1966 A N A L M O S T L I N E A R R E C U R R E N C E 121 w h e r e $ i s t w i c e the h i g h e s t p o w e r of 2 d i v i d i g k. T h e r e f o r e w h e r e if J J ^ - ( 1 - X ) ( X + 8 X X X X ^ x k " X + 8 e. ) _b (x) ^k ' a, a 1 r k = , a, > a» >... > a > 0, 1 Z r - k - 1 the c o e f f i c i e t of x i the p o w e r s e r i e s o the r i g h t h a d s i d e is a, a ^ k = e = a, 2 k a 2 r r + 2k. (The r e a d e r w i l l fid the v e r i f i c a t i o of t h i s l a t t e r f o r m u l a a i t e r - e s t i g e x e r c i s e i the u s e of the b i a r y s y s t e m. ) We c a e s t i m a t e the m a g i t u d e of w, a s f o l l o w s : a - 1 a A a k + 2k - (2 x x + " a + ' a l ' a +1 = (a +2)k a + 2 > (1+log k)k - 2k ; h e c e (7) k l o g 2 k - k 4 i i k 4 k l o g 2 k + 2k. T h i s e s t i m a t e ad the m o o t o i c i t y of <t> a r e the oly f a c t s a b o u t F(x) 'which a r e u s e d i the d e r i v a t i o b e l o w. T h e Let E - * < l ( l - x ) ) 2 G(x) = e G'(x) _ - l o g ( l - x ) _ _ u G(x) I 2 (1-x) X ' T H T X + 2 I 2 x 3 I 2 x I 2 x r - > Sice the d e r i v a t i v e of - l o g ( l - x ) / ( l -x) i s ( l - l o g ( l - x ) ) / ( l - x ), we k - 1 fid t h a t the. c o e f f i c i e t of x i the p o w e r s e r i e s o t h e r i g h t i s w h e r e

6 122 AN ALMOST LINEAR RECURRENCE April (9) h k = l + A Sice L = I k + 0(1), we have therefore established the equatios k=l k=l ad ( i i ) ^ k - x k + o(k). This suggests a p o s s i b l e relatio betwee the coefficiets of F(x) those of G(x). Note that if ad ^ ) = ( l - x ) f ( x ). the x F(x) = exp J (l-t)f(t)dt. 0 Therefore the followig lemma shows how relatios (10) ad (11) might be applied to our problem: Lemma 1. Let x A(x) = exp / (l-t)a(t)dt, 0 x B(x) = exp J (l-t)b(t)dt, where A(x) = A x, a(x) 2}k xk_1 ' B(x) 53k xk - b(x) ^}y* k ~ l A s s u m e the coefficiets of A(x) ad of b(x) a r e o-egative ad odecreasig. The if a, <_ b for all k, A, <^ B ; if a, > b for all k, A k > B k.

7 1966 AN ALMOST LINEAR RECURRENCE 12: Proof: A Q = B Q = 1. Assume a k < b R for all k, ad A < B fo 0 L k <, The sice A'(x) = (1-x)a(x)A(x), we have A = a A + a. (A -A ) a. (A. - A J) o - l x 1 o' l v ^ < b A o + b - l. x (A.-A 1 o' ) b. l v (A -1,-A -20) 7 - A (B -b J + A ^ b _-b J A. b, o < B (b -b.) + B. (b. -b 0) B. b, = B o -1 l v -1-2 ' -1 1 Essetially the same argumet works if a ^ b for all k e The problem is ow oe of estimatig the coefficiets of I ^ l ^ l - x ) G(x) = e Theorem 2 0 If (12) U. JL1JL I J. - JS.J ' V ^ 1 1 we have (13) Proof: (14) Fi-r s t we c = a I + 0((l )(l I )) show that 00 I (1-x) = > H x Zm^ m, =m where H m =y * ^ * 1 m-1 summed over all itegers a,,..., a. such that 1 = a. <, ad te 1 m-1 l the a. a r e distict. This follows iductively, sice the derivative I of (14) is I^V-x) _ Y" H x 11 " 1 (x-1) " Z J m 3 X

8 124 AN ALMOST LINEAR RECURRENCE April ad we have (15) x H = H, + " ^ H... m, m, - l -1 m - l, - l Turig to equatio (12), we have oo oo (16) V c x = V a m i 2 m ( l - x ) = ^ X ) ^ ( ) H 7 / ^ Z«^ m i^"^ ^^ ml 2m, =0 m=0 =0 m=0 (We defie H = 0 if m >, so the parethesized summatio is m, actually a fiite,sum for ay fixed value of. ) Our theorem relies o the e s t imates (17) (h. - h. ) m _ 1 < H L h 1 * 1 ". 1, if m <. -1 m - 1 ~ m, ~ -1 The righthad iequality is obvious, sice this is the sum Z ^ a i ' * ' m - 1 without the r e s t r i c t i o that the a's a r e distict. O the other had, give ay t e r m of we form a t e r m -1 x-1 L.-J a, a,, / 1 m - 1 m L a 1 4- b b 1 1 m - 1 belogig to H, where b, = a, - r if a, is the r - t h largest of & 6 te m, k k k j a,,,.. ^, o Thus, we d e c r e a s e the largest elemet b y l, the I 1 m - l. f secod largest by 2, ad so o; i case of ties, a a r b i t r a r y order is take. No two t e r m s map ito the same 1 a."* a 1 m - 1

9 1966 A N A L M O S T L I N E A R R E C U R R E N C E 125 ad 1 m m m - 1 so the lefthad s i d e of (17) is e s t a b l i s h e d. P u t t i g the r i g h t h a d s i d e of (17) ito (16), we o b t a i oo 0 JiiL m - L 2 m m 2 ah 1 T"^1 a h. -. ah (18) c ^ V ^ H, < 2 -. I Tv = e " Z_^ ( m - l ) I 2 m, ~ m = 0 O the o t h e r h a d,? m a M > H 9 A * ( m - l ). m = l (19) c > -, ( m - 1 )I 2 m, for a y p a r t i c u l a r v a l u e of m We c h o o s e m to be a p p r o x i m a t e l y 2 ah + 1, a s s u m i g is l a r g e. T h e we e v a l u a t e the l o g a r i t h m of the t e r m o the r i g h t, u s i g S t i r l i g ' s a p p r o x i m a t i o ad the left h a d s i d e of (17), ad d i s c a r d i g t e r m s of o r d e r l e s s t h a (I )(l I ): i, / 2a a /,, 2 m I c > I I T r r r ( h i " h o i ) / x I ( m - l ) I m - l / = ah, I a + 2 a h. l(h, - h 0. ) - a h, (l(ah, ) - 1) + O(l) m - l ' - l x - 1 ' = a h 2 + 2a h 2 2. l(l - m " 1 ) + 0(l ) h, ' - 1 = a h a h. h (l ) m - l T h i s t o g e t h e r w i t h (18) e s t a b l i s h e s t h e o r e m 2 0 T h e o r e m 3. Let c be a s i t h e o r e m 2. T h e,. +1. h m = 1. c *-+~ oo P r o o f : Sice H., > H, we h a v e m, +1 m, by (16). +1 y c +1

10 126 AN ALMOST LINEAR RECURRENCE April We also observe that H < h. H, ad hece by (15) m, "" -1 m - 1, thus H ^ < H + H l L h.h 0 ; m, +1 - m, -1 m-z, E a. (2m. 2a V"^ a 2m-l ) ( 2(m-l) l)!7m-2 / v ' 2(m-l), m=l ml W r ) Z m, (+1) -1 m=2 ^i I 7 3 ah. -1 < c + -ps c ~ Corollary 3. If P(x) is ay polyomial, ad if y ^ c x = e a l 2 ( l - x ) + P(x) the P(x) a Q + a x x + a 2 x I C = I c +0(1). C a 0 C + a l C - l + ' - + a 0 P(l) = _> e c c 1 2 Theorem 4. I 0 ^ -p ^-(l ) I 4 Proof: Let > 0 be give. By (11), we ca fid N so that whe > N, (1- )X < {p < (I + ) x k. Apply lemma 1 with A(x) = F(x), oo b(x)= ^ + ^ x V^1 + X ) (1+ )X l + c j xx ^ 1. k=n+l We fid <t> < C where, by Corollary 3, "~ The apply lemma 1 with I C ^ t - r i. l4' A(x) = F(x), b(x) = ^ (l- )X k x k _ 1 k=n+l

11 1966 AN ALMOST LINEAR RECURRENCE 127 This gives us <p > C' where Therefore 1-2 I C ~* ( T /l )l I 4' I 0, 1 (I ) is arbitrarily small whe is large eough. Of course, the estimate we have derived i this theorem is verycrude as far as the actual value of <t> is cocered. Empirical ^ tests based o the exact values of 4> for < reveal excel ~" let agreemet with the followig formula: (20) I < & J2-2L ( l. 2(l I ) + 1) + I I 4 The error is less tha.05 for > 10; it reaches a low of about - o 05 whe is ear 50, the icreases to approximately. 032 whe is ear 5000, ad it slowly decreases after that. Thus we ca use (20) to calculate (21) 0 ^ s472 le 12l ( 1 lil) i ' log 2 with a error of at most 5% whe 10 < < Although formula (20) gives very good accuracy, it should be remembered that oly the first term of the expasio has bee verified, ad the comparatively small values of l l for the rage of cosidered makes it possible that (20) is ot the true asymptotic result. O the assumptio that the true formula is a relatively "simple" oe, however, equatio (20) gives strikig agreemet. A similar situatio exists i the study of the partitio fuctio; the methods used here ca be applied with ease to that problem, to give log p() *v 7r \/-^ ; the actual asymptotic formula for p() itself is

12 128 A N A L M O S T L I N E A R R E C U R R E N C E A p r i l W! -I6 A r- P() - ( j ) T + 0(e V ) 4VT 4 w V r ( - ^ ) ( - l 4 ) w h e r e A < ^ " / p() <- e 4 V 3 It i s doubtful t h a t it w o u l d h a v e b e e g u e s s e d e m p i r i c a l l y i e i t h e r of t h e s e formso F o r a a c c o u t of t h i s a d a b i b l i o g r a p h y, s e e f l l. REFERENCES 1. Go H. H a r d y, R a m a u j a, c h a p t e r 8. C h e l s e a, N. Y., o K, Kopp ad I. S c h u r, E L e m e t a r e r B e w e i s e i i g e r a s y m p t o t i s c h e r F o r m e l d e r a d d i t i v e Z a h l e t h e o r i e. M a t h, Z e i t s c h r i f t 24 ( ) xxxxxxxxxxxxxxx N O T I C E T O A L L S U B S C R I B E R S! i I P l e a s e otify t h e M a a g i g E d i t o r A T O N C E of a y a d d r e s s c h a g e. T h e P o s t Office D e p a r t m e t, r a t h e r t h a f o r w a r d i g m a g a z i e s m a i l e d t h i r d c l a s s, s e d s t h e m d i r e c t l y to the d e a d - l e t t e r office. U l e s s t h e a d d r e s s e e s p e c i f i c a l l y r e q u e s t s the F i b o a c c i Q u a r t e r l y be f o r w a r d e d a t f i r s t c l a s s r a t e s to the e w a d d r e s s, he w i l l ot r e c e i v e i t. ( T h i s w i l l u s u a l l y c o s t a b o u t 30 c e t s for f i r s t - c l a s s p o s t a g e. ) If p o s s i b l e, p l e a s e otify us A T L E A S T T H R E E W E E K S P R I O R to p u b l i c a t i o d a t e s : F e b r u a r y 15, A p r i l 15, O c t o b e r 15, a d D e c e m b e r 1 5.