Accepted in Fibonacci Quarterly (2007) Archived in SEQUENCE BALANCING AND COBALANCING NUMBERS

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1 Accepted i Fiboacci Quarterly (007) Archived i SEQUENCE BALANCING AND COBALANCING NUMBERS G. K. Pada Departmet of Mathematics Natioal Istitute of Techology Rourela Orissa, Idia gpada_it@rediffmail.com Abstract: The cocept of balacig ad cobalacig umbers is geeralized to a arbitrary sequece; thereby sequece balacig umbers ad sequece cobalacig umbers are itroduced ad defied. It is proved that there does ot exist ay sequece balacig umber i the Fiboacci sequece ad the oly sequece cobalacig umber i the Fiboacci sequece is F =. Higher order balacig ad cobalacig umbers are also itroduced. A result o oexistece of third order balacig ad cobalacig umbers is also proved. A cojecture o the oexistece of solutios of higher order balacig ad cobalacig umbers is also stated at the ed. Key words: Triagular umber, Proic umber, Balacig umber, Cobalacig umber. INTRODUCTION Behera ad Pada [] defied balacig umbers as solutios of the Diophatie equatio ++ +( ) = (+) + (+) + + (+r), callig r the balacer correspodig to. They also established may importat results o balacig umbers. Later o, Pada [] idetified may beautiful properties of balacig umbers, some of which are equivalet to the correspodig results o Fiboacci umbers, ad some others are more iterestig tha the correspodig results o Fiboacci umbers. Subsequetly, Liptai [7] added aother iterestig result to the theory of balacig umbers by provig that the oly balacig umber i the Fiboacci sequece is. Behera ad Pada [] proved that the square of ay balacig umber is a triagular umber. It is also true that if r is a balacer, the r + r is a triagular umber. Subramaiam [4, 5] explored may iterestig properties of square triagular umbers without liig them to balacig umbers because of their uavailability i the literature at that time. I [6] he itroduced the cocept of almost square triagular umbers (triagular

2 umbers that differ from a square by uity) ad lied them with the square triagular umbers. Pada ad Ray [] itroduced cobalacig umbers as solutios of the Diophatie equatio ++ + = (+) + (+) + + (+r) callig r Z + the cobalacer correspodig to. The cobalacig umbers are lied to a third category of triagular umbers that are expressible as the product of two cosecutive atural umbers (approximately as the arithmetic mea of squares of two cosecutive atural umbers i.e. [ + ( + ) ]/ ( + ) ). The defiitios of balacig ad cobalacig umbers are closely related to the sequece of atural umbers. I what follows, we defie sequece balacig ad cobalacig umbers, i which the sequece of atural umbers is replaced by a arbitrary sequece of real umbers. Let { a } = be a sequece of real umbers. We call a umber a m of this sequece a sequece balacig umber if a + a am = am + am am + r for some atural umber r. Similarly, we call a m a sequece cobalacig umber if a + a am = am + am am + r for some atural umber r. For example, if we tae a = the the sequece balacig umbers of this sequece are, 70, 408, which are twice the sequece of balacig umbers, ad the sequece cobalacig umbers are 4, 8, 68, which are twice the sequece of cobalacig umbers. Similarly, if we tae a = / the the sequece balacig umbers of this sequece are, 7.5, 0, which are half the sequece of balacig umbers, ad the sequece cobalacig umbers are, 7, 4, which are half the sequece of cobalacig umbers.. SEQUENCE BALANCING AND COBALANCING NUMBERS IN CERTAIN SEQUENCES I this sectio we ivestigate sequece balacig ad cobalacig umbers i some umber sequeces. Throughout this sectio B is the th balacig umber, R is the th balacer, b is the th cobalacig umber ad r is the th cobalacer, where Z +.. Sequece balacig ad cobalacig umbers i the sequece of odd atural umbers. Let a =. The ay sequece balacig umber m of this sequece satisfies (m ) = (m ) + (m+ ) (( m+ s) )

3 for some atural umber s. This is equivalet to ( m ) + m = ( m+ s), which is a particular case of the Pythagorea equatio. Puttig y = m+ s we see that the above equatio reduces to m( m ) y y =. y y + m( m ) Sice ad must be cosecutive itegers it follows that y is a proic triagular umber. Hece, must be a cobalacig umber ([], 89). Puttig b = y we see that y = b+ ad cosequetly + 8b + 8b m =. Sice the cobalacig umbers b ad cobalacers r are related by (b ) + 8b + 8b r =, it follows that m = r + b is the required sequece balacig umber. For example for b =, r =, m = r + b = 7 ad we have + + 5= 9; similarly for b = 4, r = 6, m = r + b = 4 ad we have = Thus the sequece of sequece balacig umbers i the sequece of + r + = odd atural umbers is give by { }. b Ideed we ca express these sequece balacig umbers i terms of the balacig umbers. For this, we eed the followig results. THEOREM.. ( [], p.98). For =,,, (B ) + 8B R =. THEOREM.. ( [], p.0). For =,,, B = B 8B. + THEOREM.. ([], p.96). For =,,, R b ad r = B. = + We are ow i a positio to prove that the sequece of sequece balacig umbers i the sequece of odd atural umbers is also give by the more coveiet form { B } = B. + + THEOREM..4. The sequece of sequece balacig umbers i the sequece of odd atural umbers is give by { B B } = b + B for =,,. + r = B+ PROOF. For =,, we have + + +, i.e.,

4 ad thus, B B B = B = r This completes the proof. B (B ) + + = = R, + B = B + b + R. 8B 8B + Let the th sequece balacig umber i the sequece of odd atural umbers be deoted by x. The x ca be more coveietly calculated by a recurrece relatio. THEOREM..5. The sequece { } = x for. = 6x x x satisfies the recurrece relatio PROOF. Sice the sequece of balacig umbers satisfies the recurrece relatio B = 6B B for ( [], p.0), ad x = B+ + B, it follows that { x } = satisfies the same recurrece relatio as that of { } = B. We ext ivestigate the existece of sequece cobalacig umbers i this sequece. Ay sequece cobalacig umber m of this sequece satisfies (m ) = (m+ ) + (m+ ) (( m+ s) ) for some atural umber s. This is equivalet to m = ( m+ s), which is impossible sice is ot a square. Hece, we have the followig importat result. THEOREM..6. There does ot exist ay sequece cobalacig umbers i the sequece of odd atural umbers.. Sequece balacig ad cobalacig umbers i the sequece a = +. Ay sequece balacig umber m + of this sequece satisfies m= ( m+ ) + ( m+ ) ( m+ s ) for some atural umber s. Puttig y = m+ s we see that the above equatio is equivalet to y ( y ) = ( m+ ), y( y ) showig that is a triagular umber differig from a square by. Such a triagular umber is called a almost square triagular umber 4

5 (ASTN) [6]. Ideed, a triagular umber T for which T+ is a perfect square y( y ) is called a β-astn. Thus, is a β-astn. Let { β } = be the sequece of β-astn s. The the followig theorem gives the totality of β-astn s. THEOREM.. ([6], p.96, []). β ( B 4B ) ad = ( B+ B ) β. = + Now puttig y ( y ) β = = ( m+ ) we see that m = β+. Thus, a simple use of Theorem.. yields the followig result. THEOREM... If z deotes the th sequece balacig umber of the sequece a = +, the z = B+ 4B ad z = B B for =,,. Thus B 4B = = satisfies = , ad B B = 5 6 = satisfies = ad so o. We ext ivestigate the existece of sequece cobalacig umbers i this sequece. Ay sequece cobalacig umber m of this sequece satisfies ( m+ ) = ( m+ ) + ( m+ ) ( m+ s ) for some atural umber s, ad oce agai puttig y = m+ s we see that the above equatio is equivalet to y( y ) ( m )( m+ ) =. Thus, we must search for those proic umbers that are more tha triagular umbers. Oe such umber is 56 sice 0 56 = 7 8=. Hece, m = 7 is a sequece cobalacig umber i this sequece ad we have = Agai sice = 4 44=, it follows that m = 4 is also a sequece cobalacig umber i this sequece ad we have = Sequece balacig ad cobalacig umbers i the Fiboacci sequece. A sequece balacig umber F m i the Fiboacci sequece would satisfy 5

6 F + F = Fm + Fm + + s for some s. But it is well ow that F + F F m = Fm so that F + F F < m Fm for each atural umber m. Hece, there does ot exist ay sequece balacig umber i the Fiboacci sequece. Similarly, a sequece cobalacig umber F m i the Fiboacci sequece would satisfy F + F = Fm + Fm + + s for some s. I view of F m < F + F < Fm + Fm + for m >, it follows that o Fiboacci umber F for > ca be a sequece balacig umber. For, we have F + F = + = = F. Hece, the oly sequece balacig umber i the Fiboacci sequece is F =. The above discussio proves the followig theorems. THEOREM... There does ot exist ay sequece balacig umber i the Fiboacci sequece. THEOREM... The oly sequece cobalacig umber i the Fiboacci sequece is F =.. HIGHER ORDER BALANCING AND COBALANCING NUMBERS Let be ay atural umber. We call the sequece balacig umbers of the sequece { } = a defied by a =, the balacig umbers of order. Similarly, we call the sequece cobalacig umbers of this sequece, the cobalacig umbers of order. Thus, balacig ad cobalacig umbers of order oe are the usual balacig ad cobalacig umbers, respectively. We also call a balacig umber of order two a balacig square ad a balacig umber of order three a balacig cube. Similarly, we also call a cobalacig umber of order two a cobalacig square ad a cobalacig umber of order three a cobalacig cube. We first prove the followig result o balacig cubes ad cobalacig cubes. 6

7 THEOREM.. There does ot exist ay balacig cube or cobalacig cube. We eed the followig theorem to prove Theorem.. THEOREM. ([8], p. 77). The oly solutios of the Diophatie equatio x( x ) y( y ) = i positive itegers are ( x, y) = (,), (, ), (4,9). PROOF OF THEOREM.. Ay balacig cube must satisfy ( ) = ( + ) + ( + ) ( + r) for some atural umber r, which is equivalet to m( m) ( ) = (where m = + r). Now by Theorem., the oly possible solutios of this equatio are (m+, +) = (, ), (, ) ad (4, 9). m + = ad += implies m = = 0 which is ot possible sice m > > 0. Agai m + = ad += implies m = = which is ot possible sice m>. Lastly, m + = 4 ad += 9 implies m = ad = 8, which is agai impossible. Hece, o balacig cube exists. If is a cobalacig cube, the it satisfies = ( ) + ( + ) ( + r) for some atural umber r, which is equivalet to ( + r)( + r ) ( ) = which has o solutio i positive itegers sice is ot a square. Hece, o cobalacig cube exists. I coectio with the higher order balacig ad cobalacig umbers, the author, after exhaustive verificatio of special cases feels that the followig is true. CONJECTURE.. There exists o balacig umber or cobalacig umber of order for. More precisely, the Diophatie equatios ( ) = ( ) + ( + ) ( + r) ad = ( + ) + ( + ) ( + r) have o solutios i (, r) i positive itegers if. 4. CONCLUSION 7

8 The wor o higher order cobalacig umbers is related to some classical usolved problem i Diophatie equatios. I this cotext we recall the wors of Berstei (see [], [] ad [4]) o pyramidal Diophatie equatios. These wors, i tur, are particular cases of a problem due to Ërdös [6], amely whether the Diophatie equatio m( m)( m+ )...( m+ ) = ( + )( + )...( + ) has ay solutio for > ad m + <. Maowsi [9] aswered Ërdös questio i the egative for a particular case with the use of results of Segal []. The existece of cobalacig squares is equivalet to the existece of solutio to the Diophatie equatio m( m)( m+ ) = ( + )( + ), which is a particular case of the previous Diophatie equatio. Mordell [0] looed at particular cases of early pyramidal umbers (i.e. ay umber differig from a pyramidal umber by ) as did Boyd ad Kisilevsy [5], but the scope of geeralizatio is wide ope. Acowledgemet: It is a pleasure to tha the aoymous referee ad Professor Curtis Cooper, Editor, The Fiboacci Quarterly for their valuable commets ad suggestios that greatly improved the presetatio of the paper. REFERENCES [] A. Behera ad G. K. Pada, O the square roots of triagular umbers, The Fiboacci Quarterly, 7()(999), [] L. Berstei, The geeralized Pellia equatio, Trasactios of the America Mathematical Society, 7(968), [] L. Berstei, The liear Diophatie equatio i variables ad its applicatio, The Fiboacci Quarterly, 6(968), 6. [4] L. Berstei, Explicit solutios of pyramidal Diophatie equatios, Caadia Mathematical Bulleti, 5 (97), [5] D.W. Boyd, ad H.H. Kiisilevsy, The Diophatie equatio u ( u )( u + )( u + ) = v( v )( v + ), Pacific Joural of Mathematics, 40 (97),. [6] P. Ërdös, Problem 468, The America Mathematical Mothly, 6 (955), [7] K. Liptai, Fiboacci balacig umbers, The Fiboacci Quarterly, 4(4) (004), 0 40 [8] M. Luo, O the Diophatie equatio (x(x )/)^= (y(y )/), The Fiboacci Quarterly, 4() (996), [9] A. Maowsi, Problem 468, The America Mathematical Mothly, 70 (96), 05. 8

9 [0] L. J. Mordell, Iteger solutios of the equatio ey = ax + bx + cx+ d, Proceedigs of the Lodo Mathematical Society, (9), [] G. K. Pada ad P. K. Ray, Cobalacig umbers ad cobalacers, Iteratioal Joural of Mathematics ad Mathematical Scieces, 005(8), (005), [] G. K. Pada, Some fasciatig properties of balacig umbers, to appear i Applicatios of Fiboacci Numbers Vol. 0, Kluwer Academic Pub., 006. [] S.L. Segal, A ote o pyramidal umbers, The America Mathematical Mothly, 69 (96), [4] K. B. Subramaiam, A simple computatio of square triagular umbers, Iteratioal Joural of Mathematics Educatio i Sciece ad Techology, (5) (99), [5] K. B. Subramaiam, A divisibility property of square triagular umbers, Iteratioal Joural of Mathematics Educatio i Sciece ad Techology, 6() (995), [6] K. B. Subramaiam, Almost square triagular umbers, The Fiboacci Quarterly, 7() (999), AMS Classificatio Numbers (000): B 9, D 4 9