BINOMIAL PREDICTORS. + 2 j 1. Then n + 1 = The row of the binomial coefficients { ( n

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1 BINOMIAL PREDICTORS VLADIMIR SHEVELEV arxiv: v2 [math.nt] 22 Jul 2009 Abstract. For oegative itegers, k, cosider the set A,k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2 j0 + 2 j1 +..., j 0 > j 1 >.... The the umber is called a biomial predictor, if A,k = 2 j k, k = 0, 1,.... We give a full descriptio of the sequece of the biomial predictors. 1. Itroductio Defiitio 1. For oegative itegers, k, cosider the set A, k = { [0, 1,..., ] : 2 k ( ). Let the biary epasio of + 1 be: + 1 = 2 j j , j 0 > j 1 >.... The the umber is called a biomial predictor, if A, k = 2 j k, k = 0, 1,.... Eample 1. It is easy to see that = 0 is a biomial predictor. Ideed, A 0, 0 = {0, A 0, 0 = 2 0 that is the biary epasio of 1. Eample 2. Let = 11. The + 1 = The row of the biomial coefficiets { ( ), = 0, 1,..., 11 is: 1, 11, 55, 165, 330, 462, 462, 330,165,55, 11, 1. Here A 11,0 = 8, A 11,1 = 4. Thus, by the defiitio, 11 is a biomial predictor. Our aim is to give a full descriptio of the sequece of the biomial predictors. Below we use the followig well kow result of J.Glaisher (1899; i [1] A.Graville gives a ew elegat proof; geeralizatios see i [1]-[2]): Lemma 1. (1) A,0 = 2 s(), where s() is the umber of 1 s i the biary epasio of. Defiitio 2. A oegative iteger is called a Zumkeller s umber (see sequece A i [5]), if it has ot more tha 1 zeros i its biary epasio. Our result is the followig.

2 BINOMIAL PREDICTORS 2 Theorem 1. 0 is a biomial predictor if ad oly if it is a Zumkeller s umber. 2. Proof of ecessity A proof of ecessity we easily derive from oly the first coditio of Defiitio 1, i.e. from the equality A,0 = 2 j 0. From this equality ad Lemma 1 we coclude that (2) j 0 = s(). If ow is either a odd umber with a positive umber of 0 s i its biary epasio, or a eve umber, the the addig of 1 does ot chage the umber of biary digits, ad it equals to j 0 + 1, i.e., by (2), s() + 1. Cosequetly, the umber has eactly oe zero. We also did ot cosider a case whe a odd umber has i its biary epasio oly 1 s. Thus we proved that a biomial predictor has ot more tha oe zero, i.e. is a Zumkeller s umber. Let a() be such epoet that 3. Proof of sufficiecy (3) 2 a(). Lemma 2. (J. C. Pushta, J. Spilker [4], p.213, g = 2) For 0, we have (4) s() = i 1 /2 i. Note that this formula just a little later was obtaied idepedetly by B.Cloitre (see his commet to A i [5]). Corollary 1. (5) a(!) = s(). Proof. Ideed, as is well kow, a(!) = i 1 /2 i, ad (5) follows directly from Lemma 2. J.-P. Allouche iformed the author (private commuicatio), that as early as 1830, A.-M. Legedre [3, p.12] empirically oticed that (i our otatios) a(!) = s(); his empirical formula gives also a geeralizatio o the case of a base of prime p ad for g = p coicides with the formulatio of

3 BINOMIAL PREDICTORS 3 Lemma 4(part 1) i [4, p.213]. Astoishigly, that either Legere or Pushta ad Spilker [4] did ot discuss a simple, but, i our opiio, very importat corollary of (4)-(5) for the biomial coefficiets. We have ot foud ay discus of this i ay moder book. Theorem 2. ( ) (6) a( ) = s() + s( ) s(). Proof. Accordig to (5), we have ( ) a( ) = a(!) a(!) a(( )!) = ( s()) ( s()) ( s( ) = s() + s( ) s(). I particular, sice s(2) = s(), the for the cetral biomial coefficiets we fid ( ) ( ) a( ) = s(); a( ) = s( + 1) 1. Eample 3. For ( ) 33 we have ( ) 33 a( ) = s() + s(19) s(33) = = 4. Ideed, ( ) 33 = = Corollary 2.. For every lattice pair (, y) (0, 0), we have the triagle iequality: s( + y) s() + s(y). The equality attais if ad oly if ( +y ) is odd. Remark 1. It is iterestig to to picture o the plae the set of lattice poits {(, y) (0, 0) : ( ) +y is odd. Furthermore, we obtai the followig geeralizatio of Theorem 2 o multiomial coefficiets. Corollary 3. If = k, the we have ( ) k a( ) = s( 1, 2,..., i ) s(). k Corollary 4.. The umber of solutios of the equatio i=1

4 BINOMIAL PREDICTORS 4 (7) s() + s( ) = s(), [0, 1,..., ], equals to 2 s(). Proof. The corollary immediately follows from Theorem 2 ad Lemma 1. the umber of solutios of the equa- Furthermore, for r 0, deote λ (r) tio (8) s() + s( ) = s() + r, [0, 1,..., ]. I particular, from Corollary 4 it follows that (9) A,0 = λ (0) = 2s(). Let be a Zumkeller s umber. Cosider, first, a trivial case whe = 2 t 1 = The (7) evidetly satisfies for every [0, 1,..., ] ad, t by Lemma 1, we have λ (0) = 2 t = + 1, i.e. is a biomial predictor. Let ow have a uique zero i its biary epasio. Cosider such of the geeral form: (10) = , t u such that the first digit of is 1. The + 1 = = 2 u+t + 2 u+t u = t+1 u (11) 2 s() + 2 s() s() t. For [0, 1,..., ] distiguish two cases: 1) has the form (12) = 0 t u where deotes arbitrary biary digit; 2) has the form (13) = t v v t,. u Sice, the it is clear that i this case the legth of is less tha. It easy to see that i Case 1) the umber of differet values of [0, 1,..., ] is 2 t+u = 2 s() ad all of these values satisfy equatio (7), i.e. are i A,0. Therefore, i view of (9), Case 1 ehausts all values satisfyig (7). Now we cosider Case 2. Subtractig (13) from (10), we have

5 BINOMIAL PREDICTORS 5 () = (1 )...(1 ) (1 )...(1 ). t v v u Thus, from (13)-() we fid s( ) + s() = (t v) + 2v + u = t + u + v = s() + v. Cosequetly, of the form (13) is a solutio of equatio (8) for r = v. Hece, as it easily to see from (13), the umber of the differet Zumkeller s solutios of equatio (8) for r = v equals to 2 t v 2 u = 2 s() v, v = 1,..., t, which together with solutios of (7) correspods to (11). This completes the proof of Theorem 1. Ackowledgmet. The author is grateful to J.-P. Allouche for sedig a scaed copy of pp of Legedre book [3]. Refereces [1]. A Graville, Zaphod Beeblebro s brai ad the fifty-ith row of Pascal s triagle, Amer. Math. Mothly, 99, o. 4 (1992), ; 104, o. 9 (1997), [2]. J. G. Huard, B. K. Spearma, K. S. Williams, Pascal s triagle (mod 8), Europ. J. Combi., 19, o.1 (1998), [3]. A.-M. Legedre, Théorie de Nombres, Firmi Didot Fréres, Paris, [4]. J.-C. Pushta, J. Spilker,Altes ud eues zur QuersummeMath. Semesterber. 49 (2002), [5]. N. J. A. Sloae, The O-Lie Ecyclopedia of Iteger Sequeces (http: // Departmets of Mathematics, Be-Gurio Uiversity of the Negev, Beer- Sheva 84105, Israel. shevelev@bgu.ac.il