#A INTEGERS 5A (05) THE SIGNUM EQUATION FOR ERDŐS-SURÁNYI SEQUENCES Doin Andica Faculty of Mathematics and Comute Science, Babeş-Bolyai Univesity, Cluj-Naoca, Romania dandica@math.ubbcluj.o Eugen J. Ionascu Deatment of Mathematics, Columbus State Univesity, Columbus, Geogia ionascu@columbusstate.edu Received: 3//4, Revised: 3/3/5, Acceted: 5/9/5, Published: 6/5/5 Abstact Fo an Edős-Suányi sequence it is customay to conside its signum equation. Based on some classical heuistic aguments, we conjectue the asymtotic behavio fo the numbe of solutions of this signum equation in the case of the sequence {n k } n (k ) and the sequence of imes. Suisingly, we show that this method does not aly at all fo the Fibonacci sequence. By comuting the ecise numbe of solutions, in this case, we obtain an exonential gowth, which shows, in aticula, the limitations of such an intuition.. INTRODUCTION We ecall (see [6] fo a n = n ) that a sequence of distinct ositive integes {a m } m is an Edős-Suányi sequence if evey intege k can be witten in the fom k = ±a ± a ± ± a n fo some choices of signs + and, in infinitely many ways. Using an induction agument, it iot di cult to see that {m} m, o even {m } m, is an Edős- Suányi sequence. In 979, J. Mitek oved in [8] that in fact, fo an abitay ositive intege s, {m s } m is an Edős-Suányi sequence. M. O. Dimbe gives ([4] and [5]) su cient conditions fo a sequence to be an Edős-Suányi sequence. THEOREM. Let {a m } m be a sequence of distinct ositive integes such that a = and fo evey n, a n+ ale a + + a n +. If the sequence {a m } m contains infinitely many odd integes, then it is an Edős-Suányi sequence. Honoific Membe of the Romanian Institute of Mathematics Simion Stoilow
INTEGERS: 5A (05) As an alication, one can check that the Fibonacci sequence satisfies the hyothesis of this theoem, since F 0 + F + + F n = F n+, n. Since F n + F n+ F n+ = 0 fo n 0, it iot suising that k = ±F ± F ± ± F n has infinitely many eesentations if it has one. The existence of at least one eesentation is elated to Zeckendof s theoem (see []) and as a cuiosity we include one such eesentation fo k = 05: 05 = 9X j F j, with j =, if and only if j {4,, 5, 7, 9}. THEOREM. Let f Q[X] be a olynomial such that fo any n Z, f(n) is an intege. If the geatest common facto of the tems of the sequence {f(n)} n is equal to, then {f(n)} n is an Edős-Suányi sequence. Using this esult, one can obtain many Edős-Suányi sequences: {(an ) k } n fo fixed a, k N, o { n+s s } n fo fixed s, ae just some ossible examles. We found that it is di cult to use an induction tye agument to show that these ae Edős-Suányi sequences. In this ae, given an Edős-Suányi sequence a = {a m } m, we ae inteested in the signum equation of a at level n, that is ±a ± a ± ± a n = 0. () Fo a fixed intege n, a solution to the signum equation is a choice of signs + and that makes () tue. We denote by S a (n) the numbe of solutions of Equation (). Clealy, if doeot divide u n, whee u n = a + + a n, then we have S a (n) = 0. We include next a few othe oeties of the sequence S a (n) (see [], []). (i) S a (n)/ n is the unique eal numbe having the oety that the function f : R! R, defined by 8 < cos a a x cos x f(x) = cos an x if x 6= 0, : if x = 0, is a deivative, i.e., it is the deivative of a di eentiable function on R. (ii) S a (n) is the tem not deending on z in the exansion of (z a + z a )(za + z ) a (zan + z an ). (iii) S a (n) is the coe cient of z un/ in the olynomial ( + z a )( + z a ) ( + z an ).
INTEGERS: 5A (05) 3 (iv) S a (n) can be calculated using the following integal fomula S a (n) = n Z 0 cos(a t) cos(a t) cos(a n t)dt. () (v) S a (n) is the numbe of odeed biatitions of the set {a, a,, a n } into two classes having equal sums. (vi) S a (n) is the numbe of distinct subsets of {a, a,, a n } whose elements sum to u n / if divides u n, and S a (n) = 0 othewise. (vii) S a (n) = N a (n), whee N a (n) is the numbe of eesentations of a n as a n = ±a ± a ± ± a n. In the next section, we ae inteested in the asymtotic behavio of S a (n), when n!.. Heuistic Asymtotic Behavio fo S a (n) S. Finch [7] has a vey inteesting obabilistic but heuistic aoach to aive at asymtotic esults fo S a (n). In this section, we ae going to descibe it in detail stating with an Edős-Suányi sequence a. Fo a ositive intege j, we conside the andom vaiable E j taking two values: a j o a j with obability / each. We assume that fo i 6= j, each E i is indeendent of E j. We then let the andom vaiable S n := E + E + + E n. (3) Since each of the E j has exectation zeo, we see that the exectation of S n is E(S n ) = P n E(E j) = 0. Because each of the E j has vaiation j with j = Va(E j ) = E( E j E(E j ) ) = a j, and the andom vaiables ae mutually indeendent, we conclude that s n = Va(S n ) = Va(E j ) = a j. We would like to use the Bey-Essen Theoem ([3]), which is a vey good vesion of the Cental Limit Theoem fo this situation. THEOREM. Let X, X,..., be indeendent andom vaiables with E(X j ) = 0, E(Xj ) = j > 0 and j = E( X j 3 ) <. If v u G n = (X + X + + X n )/ t j,
INTEGERS: 5A (05) 4 then su xr P (G n ale x) fo some univesal constant C. Z x e t dt ale C max alejalen q Pn j j j (4) THEOREM. Let a = {a n } n be a non-deceasing Edős-Suányi sequence of ositive integes, and let {S a (n)} n be the sequence counting the numbe of solutions of its signum equations. If we set s n = a + a + + a n, then n S a(n) ale C a n, (5) fo n big enough and fo some constant C (which doeot deend on a). Poof. We ae setting X j = E j in Theoem. and obseve that j = a 3 j = a n. Inequality (4) then becomes max alejalen j j su P (S n ale x ) xr In aticula fo y < x, we have P (y < S n ale x ) Z x e t a n dt ale C. Z x y e t dt ale C a n. and so Fo an abitay (0, ) and x =, y =, let us obseve that P (y < S n ale x ) = P ( < S n < ) = n S a(n) and so the inequality above can be witten as n S a(n) Z e t dt ale C a n. Since Z e t dt Z dt = letting! 0 and inceasing the constant C slightly, we get n S a(n) ale C a n.
INTEGERS: 5A (05) 5 Remak: In geneal, the constant C in (5) is the best that one can obtain, but in some aticula cases the constant C can be elaced by a sequence convegent to zeo. If this sequence is such that Ca n = o() then (5) becomes a genuine asymtotic fomula: S a (n) n = + o(). (6) We next discuss instances when it is vey likely that this is the case and an examle when this is clealy false... The Sequence n k If k N, and a = {n k } n, it is easy to show that fo > 0 we have j = n + + + O(n ). This shows that in this case = k+ n k+ + O(n k ). The asymtotic fomula (6) suggests that we can fomulate the following: CONJECTURE. Fo evey ositive intege k the following elation holds: lim n! n 0 o 3 (mod 4) S k (n) n n k+ (k + ) =, (7) whee S k (n) stands fo S a (n) in this case. Fo k = the elation (7) was stated in the ae [] and it was called the Andica-Tomescu Conjectue in the ecent ae [0]. It was ecently oved to be coect by B. Sullivan [0]. Fo k, although we do not yet have a oof, we believe that unde simila estictions on n as in the case k =, (7) is still valid... The Sequence of Pimes The sequence {a} = {} of imes =, = 3,... is an Edös-Suányi sequence. This was shown in an ingenious way in the ae [4] by using the esult in Theoem.. Using the Pime Numbe Theoem, and a esult fom ([9]), we deive the following asymtotic identity: s n = j = ( + o(n)) (ln n + ln(ln n) )3 n3 = ( + o(n)) 3 ln n 3 n3 (ln n). Since the ode of gowth of iot geate than n, Equation (6) suggests that the following may be tue.
INTEGERS: 5A (05) 6 Figue : Distibution of S, and N (0, 88) CONJECTURE. If is the sequence of imes, then S (n) lim n!, n odd n n n ln n = 6. (8) In 830, H. F. Schek conjectued the following ecusive fomulas geneating the m-th ime fom the evious m imes: fo evey ositive intege n, and n = ± ± ± ± n + n n+ = ± ± ± ± n + n, fo some choices of signs + and. Such eesentations wee shown to exist in 98 by S. S. Pillai. Accoding to oety (viii) (age 3), the numbe of eesentations of n+ as n+ = ± ± ± ± n fo some choices of signs + and is N (n + ) = S (n + ). It is easy to see that S (n) = 0 fo evey even n and fo instance S () = 0, S (3) =, S (5) = ( 3+5+7 = 0),..., which is ecisely half the sequence A0894 in the Online Encycloedia of Intege Sequences. Using the fist 000 tems, we calculated that S (n) n n (ln n + ln(ln n) )3/ 6 n.3897 fo n odd, n > 99, ln n which suots Conjectue. In Figue, we have included the distibution of X := ± ± ± which has zeo mean and a standad deviation of about 87.98. Hee X takes even values fom 7 to 7. As we can see, this gah di es clealy fom that of the dilated nomal distibution N (0, 88) but if we account fo the half the values, the odd values whee the obability is zeo, we see an almost efect matching with N (0, 88) (Figue ) (the facto of which shows u in the oof of Theoem.3).
INTEGERS: 5A (05) 7 Figue : Distibution of S, and N (0, 88).3. The Fibonacci Sequence In contast, let us look at the well-known sequence F 0 = 0, F = F =, F 3 =,..., F n+ = F n + F n fo n, and at the elated sequence {S F (n)} n 0, whee S F (n) is the numbe of witings of 0 as a sum ±F 0 ± F ± F ± ± F n. Fo instance, it is easy to check that S F (0) =, S F () = 0 and S F () = 4. In fact, we have the following suising fomula in geneal. THEOREM.3 Fo n = 3(k ) +, k N, and {0,, } we have 8 >< k if = 0, S F (n) = 0 if =, >: k+ if =. (9) Poof. It is known that fo evey n N, we have F 0 + F +... + F n = F n+. Fo a fixed n N, it is clea that S F (n + ) is twice the numbe of witings of F n+ as ±F n+ ± ± F ± F 0. Let us assume we have such a witing of F n+ : F n+ = n+ F n+ + n F n + + 0 F 0, with i {, }, i = 0,,,, n+. (0) We claim that that n+ = and n =. By way of contadiction, if n+ = then F n+3 = F n+ + F n+ = n F n + + 0 F 0 ale F 0 +... + F n = F n+ < F n+3. This contadiction shows that n+ =. Then Equation (0) becomes F n = F n+ F n+ = n F n + + F + 0 F 0.
INTEGERS: 5A (05) 8 Again, if n = then we obtain F n = n F n + + F + 0 F 0 ale F n+, F n ale F n ) F n < F n. So, this contadiction says that we must have n = and theefoe (0) is equivalent to 0 = n F n + F + 0 F 0, with i {, }, i = 0,,,, n. This means that S F (n + ) = S F (n ). Taking into account that S F (0) =, S F () = 0, S F () = 4, using an induction agument we see that we get the fomula in (9). It is known that fo evey n N, F0 + F + + Fn = F n F n+ and so = Fn F n+. Hence the left-hand side in (6) can be simlified to S F (n) n s n 8 8 F3k >< k 3 F 3k if n = 3k 3, = 0 if n = 3k, >: 4 F3k k F 3k if n = 3k. If we denote golden atio, as usual, by, i.e., = 5+, then since F n = n 5 +o(), we have 3k 5/ F 3k 3 F 3k = 5 +o() and F 3k F 3k = +o(). Because 3 /4.059 we see that S F (n) lim n!,n 0 o (mod 3) n =, 3k / 5 which makes (6) vey fa fom giving any asymtotic infomation about S F (n). Refeences [] Andica, D., Ionascu, E. J., Some Unexected Connections Between Analysis and Combinatoics, in Mathematics without boundaies. Toics in ue Mathematics, Th.M. Rassias and P. Padalos, Eds., Singe (04), -0 [] Andica, D., Tomescu, I., On an intege sequence elated to a oduct of tigonometic fuctions, and its combinatoial elevance, J. Intege Seq. 5(00), Aticle 0..4. [3] Bey, A. C., The Accuacy of the Gaussian Aoximation to the Sum of Indeendent Vaiates, Tans. Ame. Math. Soc. 49()(94), -36 [4] Dimbe, M. O., A oblem of eesentation of integes (Romanian), G.M.-B, 0-(983), 38-383 [5] Dimbe, M. O., Genealization of eesentation theoem of Edős and Suányi, Comment. Math. XXVII(988), No., 33-35.
INTEGERS: 5A (05) 9 [6] Edős, P., Suányi, J., Toics in the Theoy of Numbes, Singe-Velag, 003. [7] Finch, S. R., Signum equations and extemal coe cients, eole.fas.havad.edu/ sfinch/ [8] Mitek, J., Genealization of a theoem of Edős and Suányi, Comment. Math. XXI(979) [9] T. Salát, and S. Znám, On the sums of ime owes, Acta Fac. Re. Univ. Com. Math. (968), 5 [0] Sullivan, B.D., On a Conjectue of Andica and Tomescu, J. Intege Seq. 6(03), Aticle 3.3.. [] Zeckendof, E., Reésentation deombeatuels a une somme de nombes de Fibonacci ou deombes de Lucas, Bull. Soc. R. Sci. Liège 4(97), 79-8