Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 16 (2015), No. 2, pp. 1219–1231 DOI: 10.18514/MMN.2015.1724 BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS Received 22 June, 2015 Abstract. In this work, we derive some functions on balancing, cobalancing, Lucas–balancing, Lucas–cobalancing, Pell, Pell–Lucas and square triangular numbers. At the end of this article we investigated common values of combinatorial numbers and Lucas–balancing numbers. 2010 Mathematics Subject Classification: 11B37, 11B39, 11D25 Keywords: balancing numbers, Pell numbers, Pell–Lucas numbers, balancing functions, square triangular numbers 1. INTRODUCTION A positive integer n is called a balancing number (see [1] and [3]) if the Diophant- ine equation 1C2C�� �C .n�1/D .nC1/C .nC2/C�� �C .nC r/ (1.1) holds for some positive integer r which is called cobalancing number (or balan- cer). For example 6;35;204;1189 and 6930 are balancing numbers with balancers 2;14;84; 492 and 2870, respectively. If n is a balancing number with balancer r , then from (1.1) one has .n�1/n 2 D rnC r.rC1/ 2 ; and so r D �.2nC1/C p 8n2C1 2 and nD 2rC1C p 8r2C8rC1 2 : (1.2) Let Bn denote the nth balancing number, and let bn denote the nth cobalancing number. Then they satisfy B1 D 1;B2 D 6;BnC1 D 6Bn�Bn�1 and b1 D 0;b2 D 2;bnC1 D 6bn� bn�1C 2 for n � 2. The zeros of the characteristic equation x2� 6xC1D 0 for balancing numbers are ˛1 D 3C p 8 and ˇ1 D 3� p 8. Ray derived some nice results on balancing and cobalancing numbers in his Phd thesis [14, p.19]. Since x is a balancing number if and only if 8x2C 1 is a perfect square, he set y2�8x2 D 1 for some y ¤ 0, which is a Pell equation. The fundamental solution is .y1;x1/D .3;1/. So ynCxn p 8D .3C p 8/n and yn�xn p 8D .3� p 8/n for n� 1. This research was (partially) carried out in the framework of the Center of Excellence of Mechat- ronics and Logistics at the University of Miskolc. c 2015 Miskolc University Press brought to you by COREView metadata, citation and similar papers at core.ac.uk provided by Repository of the Academy's Library https://core.ac.uk/display/145235686?utm_source=pdf&utm_medium=banner&utm_campaign=pdf-decoration-v1 1220 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS Thus xn D .3C p 8/n�.3� p 8/n 2 p 8 which is the Binet formula for balancing numbers and is denoted by Bn. Let ˛D 1C p 2 and ˇD 1� p 2 be the roots of the characteristic equation for Pell and Pell–Lucas numbers defined by P0 D 0;P1 D 1;Pn D 2Pn�1CPn�2 andQ0 D Q1 D 2;Qn D 2Qn�1CQn�2 for n� 2, respectively. Notice that ˛2 D 3C p 8 and ˇ2 D 3� p 8. So the Binet formula for balancing numbers is Bn D ˛2n�ˇ2n 4 p 2 : Thus there is a connection between balancing and Pell numbers by Bn D P2n 2 . Similarly, the Binet formula for cobalancing numbers is bn D ˛2n�1�ˇ2n�1 4 p 2 � 1 2 , and so bn D P2n�1�1 2 . We note that Bn is a balancing number if and only if 8B2nC1 is a perfect square, and bn is a cobalancing number if and only if 8b2nC8bnC1 is a perfect square. Thus (1.2) implies that Cn D q 8B2nC1 and cn D q 8b2nC8bnC1 (1.3) are integers called the nth Lucas–balancing and nth Lucas–cobalancing number, re- spectively. Their Binet formulas are Cn D ˛2nCˇ2n 2 and cn D ˛2n�1Cˇ2n�1 2 , respect- ively (for further details see [9, 11–13]). Balancing numbers and their generalizations have been investigated by several au- thors, from many aspects. In [5, Theorem 4], Liptai proved that there is no Fibonacci balancing number except 1, and in [6, Theorem 4] he proved that there is no Lucas balancing number. In [15], Szalay considered the same problem and obtained some nice results by a different method. In [4], Kovács, Liptai and Olajos extended the concept of balancing numbers to the .a;b/-balancing numbers defined as follows: Let a > 0 and b � 0 be coprime integers. If .aCb/C�� �C .a.n�1/Cb/D .a.nC1/Cb/C�� �C .a.nC r/Cb/ for some positive integers n and r , then anC b is an .a;b/-balancing number. The sequence of .a;b/-balancing numbers is denoted by B.a;b/m for m� 1. In [7], Liptai, Luca, Pintér and Szalay generalized the notion of balancing numbers to numbers defined as follows: Let y;k; l 2 ZC such that y � 4. Then a positive integer x with x � y�2 is called a .k; l/-power numerical center for y if 1kC�� �C .x�1/k D .xC1/lC�� �C .y�1/l : They studied the number of solutions of the equation above and proved several ef- fective and ineffective finiteness results for .k; l/-power numerical centers. For positive integers k;x, let ˘k.x/D x.xC1/ : : : .xCk�1/: (1.4) BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS 1221 It was proved in [4, Theorem 3 and Theorem 4] that the equation Bm D ˘k.x/ for fixed integer k� 2 has only infinitely many solutions and for k 2 f2;3;4g all solutions were determined. In [17, Theorem 1] Tengely, considered the case k D 5, that is, Bm D x.xC1/.xC2/.xC3/.xC4/; and proved that this Diophantine equation has no solution for m� 0 and x 2Z. Recently in [2], Dash, Ota and Dash considered the so-called t -balancing numbers for an integer t � 2. A positive integer n is a t -balancing number if 1C2C�� �CnD .nC1C t /C .nC2C t /C�� �C .nC rC t / holds for some positive integer r which is called t -cobalancing number. The t - balancing numbers are denoted by B tn and t -cobalancing numbers by btn. In [16], Tekcan, Tayat and Özbek derived the general terms of t -balancing numbers by solv- ing the Diophantine equation 8x2�y2C8x.1C t /C .2tC1/2 D 0: Balancing functions are also interesting in the theory of balancing numbers. There were first considered by Behera, Panda and Ray in [1,14], who derived the following results. Lemma 1 ([1, Theorem 2.1]). For any balancing number x, the functions F.x/D 2x p 8x2C1, G.x/D 3xC p 8x2C1 and H.x/D 17xC6 p 8x2C1 are also balancing numbers. Lemma 2 ([1, Theorem 2.2]). If x any balancing number, then K.x/D 6x p 8x2C1C16x2C1 is an odd balancing number. Lemma 3 ([1, Theorem 4.1]). If x and y are balancing numbers, then f .x;y/D x q 8y2C1Cy p 8x2C1 is also a balancing number. Lemma 4 ([14, 2.4.12 Theorem]). If x;y and ´ are balancing numbers, then f .x;y;´/D x q 8y2C1 p 8´2C1Cy p 8x2C1 p 8´2C1 C´ p 8x2C1 q 8y2C1C8xy´ is also a balancing number. Similarly to balancing numbers, Panda and Ray [12] defined the following func- tions for any two cobalancing numbers x and y: f .x/D 3xC p 8x2C8xC1C1; g.x/D 17xC6 p 8x2C8xC1C8; 1222 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS h.x/D 8x2C8xC1C .2xC1/ p 8x2C8xC1; t.x;y/D 1 2 h 2.2xC1/.2yC1/C .2xC1/ q 8y2C8yC1 C .2yC1/ p 8x2C8xC1C p 8x2C8xC1 q 8y2C8yC1�1 i : They proved the following two lemmas. Lemma 5 ([12, Theorem 2.1]). For any two cobalancing numbers x and y, f .x/, g.x/, h.x/ and t .x;y/ are all cobalancing numbers. Lemma 6 ([12, Theorem 2.2]). If x any cobalancing number, then the cobalancing number next to x is f .x/D 3xC p 8x2C8xC1C1, and consequently, the previous one is ef .x/D 3x�p8x2C8xC1C1. Here we note, [12] originally contained a type at h.x/, we always use the correct version. 2. MAIN RESULTS In this section, we deal with balancing, Pell, Pell–Lucas and square triangular functions. Before considering balancing functions, we need the following theorem. Theorem 1. For any integers n;k; l � 0, we get BnCkCl D BkClCnCCkBlCnCCkClBnC8BkBlBn: (2.1) Proof. It is known that BnCk D CkBnCBkCn and CkCl D CkCl C 8BkBl for integers n; l;k � 0. So we easily get BkClCnCCkBlCnCCkClBnC8BkBlBn D Cn.BkClCCkBl/CBn.CkClC8BkBl/ D CnBkClCBnCkCl D BnCkCl as we wanted. � In Lemmas 1–5, Behara, Panda and Ray did not determine precisely value of the functions F.x/, G.x/, H.x/, K.x/, f .x;y/, f .x;y;´/ and f .x/, g.x/, h.x/, t .x;y/. In the following theorem, we are able to do that. Theorem 2. (1) For the balancing functionsF.x/;G.x/;H.x/;K.x/;f .x;y/; f .x;y;´/, we have F.Bn/D B2n; G.Bn/D BnC1; H.Bn/D BnC2; K.Bn/D B2nC1; f .Bn;Bk/D BnCk and f .Bn;Bk;Bl/D BnCkCl : BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS 1223 (2) For the cobalancing functions f .x/;g.x/;h.x/; t.x;y/, we have f .bn/D bnC1; g.bn/D bnC2; h.bn/D b2n and t .bn;bm/D bnCm: Proof. (1) Since p 8B2nC1D Cn, we easily get F.Bn/D 2 � ˛2n�ˇ2n 4 p 2 �� ˛2nCˇ2n 2 � D ˛4n�ˇ4n 4 p 2 D B2n: Taking k D 1 and l D 0 in (2.1), one obtains BnC1 D 3BnCCn D 3BnC q 8B2nC1DG.Bn/: If k D 2 and l D 0, then it leads to BnC2 D 17BnC6Cn D 17BnC6 q 8B2nC1DH.Bn/ and taking k D 1 and l D n, we get B2nC1 D 6BnCnC16B 2 nC1D 6Bn q 8B2nC1C16B 2 nC1DK.Bn/: Finally, if l D 0, we have BnCk D BnCkCBkCn D Bn q 8B2 k C1CBk q 8B2nC1D f .Bn;Bk/: The last assertions is obvious from (2.1) since BnCkCl D BnCkClCBkCnClCBlCnCkC8BnBkBl D Bn q 8B2 k C1 q 8B2 l C1CBk q 8B2nC1 q 8B2 l C1 CBl q 8B2nC1 q 8B2 k C1C8BnBkBl D f .Bn;Bk;Bl/: (2) It can be proved similarly. � Apart from above theorem, we can give the following theorem. Theorem 3. (1) If x is the nth balancing number and y is the nth cobalancing number, then the function B1.x;y/D �8xyC p 8x2C1 p 8y2C8yC1C1 4 is the nth balancing number, and the function B2.x;y/D xC2yC p 8x2C1C q 8y2C8yC1C1 is the .nC1/st balancing number. 1224 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS (2) (The generalized type of Lemma 2) Let Bk denote the kth balancing number, let bk denote the ktk cobalancing number and let Ck denote the ktk Lucas– balancing number for an integer k � 0. Then for any balancing number x, the function B1k.x/D Bkx p 8x2C1CCkx 2 C1C2b2nCk � n�2X iD0 B2iCkC1 is a balancing number. In fact if x is the nth balancing number, then B1 k .x/ is the .2nCk/th balancing number. So B1 k .x/ is odd balancing number for odd k and even balancing number for even k. (3) Let Bk denote the kth balancing number, let bk denote the ktk cobalancing number and let Ck denote the kth Lucas–balancing number for an integer k � 0. Then for any cobalancing number x, the function b1k.x/D CkxCBk �p 8x2C8xC1C1 � Cbk is a cobalancing number. In fact, if x is the nth cobalancing number, then bk.x/ is the .nC k/th cobalancing number (Note that b11.x/ D f .x/ and b12.x/D g.x/). (4) If x is the nth balancing number, y is the mth cobalancing number and ´ is the nth Lucas–balancing number, then the function b1.x/D 2xC p 8x2C1�1 2 is the .nC1/th cobalancing number and the function b2.x;y;´/D 2x p 8y2C8yC1C2y p 8x2C1C´�1 2 is the .nCm/th cobalancing number. Proof. (1) Notice that Bn D ˛2n�ˇ2n 4 p 2 ;bn D ˛2n�1�ˇ2n�1 4 p 2 � 1 2 ;Cn D ˛2nCˇ2n 2 and cn D ˛2n�1Cˇ2n�1 2 . So for any balancing number x and cobalancing number y, we have B1.x;y/D �8xyC p 8x2C1 p 8y2C8yC1C1 4 D �8 � ˛2n�ˇ2n 4 p 2 �� ˛2n�1�ˇ2n�1 4 p 2 � 1 2 � C � ˛2nCˇ2n 2 �� ˛2n�1Cˇ2n�1 2 � C1 4 D ( �˛4n�1C˛2nˇ2n�1Cˇ2n˛2n�1�ˇ4n�1 4 C ˛2n�ˇ2n p 2 C ˛4n�1C˛2nˇ2n�1Cˇ2n˛2n�1Cˇ4n�1 4 C1 ) 4 BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS 1225 D .˛ˇ/2n.˛�1Cˇ�1/ 2 C ˛2n�ˇ2n p 2 C1 4 D ˛2n�ˇ2n 4 p 2 D Bn since ˛ˇ D�1 and ˛�1Cˇ�1 D�2: The others can be proved similarly. � As in Theorem 3, there are infinitely many Lucas–balancing and Lucas–cobalancing functions given below which can be proved similarly. Theorem 4. Let Bk denote the kth balancing number, Ck denote the kth Lucas– balancing number and let ck denote the ktk Lucas–cobalancing number. Then for any balancing number x and any cobalancing number y with the same order, say n, (1) the function Ck.x;y/D Ck�1 p 8x2C1C ck�1 q 8y2C8yC1C cnCk�1 is the .nCk�1/st Lucas–balancing number, that is, Ck.x;y/DCnCk�1 for k � 2: (2) the function ck.x;y/D Ck�1 q 8y2C8yC1C ck�1 p 8x2C1CCn�kC1 is the .nC k� 1/st Lucas–cobalancing number, that is, ck.x;y/ D cnCk�1 for n� k�1� 1; and the function ck.y/D 4Bk.2yC1/CCk q 8y2C8yC1 is the .nCk/th Lucas–cobalancing number, that is, ck.y/D cnCk : For Pell and Pell–Lucas numbers, we can give the following theorem. Theorem 5. Let Bk denote the kth balancing number, let bk denote the kth co- balancing number, let Ck denote the ktk Lucas–balancing number, let ck denote the ktk Lucas–cobalancing number, let Pk denote the kth Pell number and let Qk denote the kth Pell–Lucas number for an integer k � 0. Then for any balancing number x, for any cobalancing number y, for any Lucas–balancing number ´ and for any Lucas–cobalancing number w with the same order, say n, (1) the function P 1k .x;y/D 8Bk.xCy/CCk. p 8x2C1C p 8y2C8yC1/ 2 CP2k 1226 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS is the .2nC2k/th Pell number, that is, P 1 k .x;y/D P2nC2k , the function P 2k .x;y/D Ck p 8x2C1C ck p 8y2C8yC1 2 is the .2nC 2k � 1/st Pell number, that is, P 2 k .x;y/ D P2nC2k�1, and the function P1.x;y;´;w/D ´ p 8x2C1Cw p 8y2C8yC1 2 is the .4n�1/st Pell number, that is, P1.x;y;´;w/D P4n�1: (2) the function Qk.x/D 32Ckx 2 C32Bkx p 8x2C1CQ2k is the .4nC2k/th Pell–Lucas number, that is, Qk.x/DQ4nC2k . (3) We set Gk.x;y/D Bk p 8x2C1Cbk q 8y2C8yC1: (a) If k D 1, then the function G1.x;y/ is the nth Lucas–balancing number, that is, G1.x;y/D Cn. (b) If k D 2, then the function G2.x;y/ is the four times of the .2nC 1/st Pell number, that is, G2.x;y/D 4P2nC1: (c) The functionGk.x;y/ is the sum of Lucas–balancing numbers from n to .nCk�1/, that is, Gk.x;y/D nCk�1X iDn Ci for every k � 1, or is the sum of Pell numbers, that is, Gk.x;y/D 8̂̂̂̂ <̂ ˆ̂̂: 4 k 2P iD1 P2nC4i�3 for even k � 2 P2nCP2n�1C4 k�3 2P iD0 P2nC4iC3 for odd k � 3: Proof. (1) Notice that CnC cn D 2P2n, BnCbn D Cn�1 2 and P2n D 2Bn. So P 1k .x;y/D 8Bk.xCy/CCk. p 8x2C1C p 8y2C8yC1/ 2 CP2k D 8Bk.BnCbn/CCk.CnC cn/ 2 C2Bk D 8Bk. Cn�1 2 /C2CkP2n 2 C2Bk BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS 1227 D 2BkCnCCkP2n D 2 ˛2k �ˇ2k 4 p 2 !� ˛2nCˇ2n 2 � C ˛2kCˇ2k 2 !� ˛2n�ˇ2n 2 p 2 � D � ˛2.kCn/C˛2kˇ2n�ˇ2k˛2n�ˇ2.nCk/ C˛2.kCn/�˛2kˇ2nCˇ2k˛2n�ˇ2.kCn/ � 4 p 2 D ˛2.nCk/�ˇ2.nCk/ 2 p 2 D P2nC2k : The others can be proved similarly. � There is a connection between balancing numbers and triangular numbers denoted by Tn which are the numbers of the form Tn D n.nC1/ 2 for n� 0. There are infinitely many triangular numbers that are also square numbers which are called square trian- gular numbers and is denoted by Sn. For the nth square triangular number Sn, we can write Sn D s2n D tn.tnC1/ 2 , where sn and tn are the sides of the corresponding square and triangle. Their Binet formulas are Sn D . ˛2n�ˇ2n 4 p 2 /2; sn D ˛2n�ˇ2n 4 p 2 and tn D ˛2nCˇ2n�2 4 : In [10], Özkoç, Tekcan and Gözeri derived some nice results on triangular, square triangular and balancing numbers. For the square triangular numbers, squares and triangles, we can give the following theorem. Theorem 6. For any balancing number x and any cobalancing number y with the same order, say n, (1) the function S.x;y/D 4x2C4y.yC1/C p 8x2C1 p 8y2C8yC1C1 8 is the nth square triangular number, that is, S.x;y/D Sn: (2) the function s.x;y/D 6y�2 p 8x2C1C3 p 8y2C8yC1C3 2 is the .n�1/st square, that is, s.x;y/D sn�1: (3) the function t .x;y/D 2.x�y�1/C p 8x2C1� p 8y2C8yC1 2 1228 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS is the nth triangle, that is, t .x;y/D tn: Proof. (1) Since 1C˛�2C2˛�1 D 1Cˇ�2C2ˇ�1 D 2 and ˛ˇ D�1, we get S.x;y/D 4x2C4y.yC1/C p 8x2C1 p 8y2C8yC1C1 8 D 8<: 4 � ˛2n�ˇ2n 4 p 2 �2 C4 � ˛2n�1�ˇ2n�1 4 p 2 � 1 2 �h ˛2n�1�ˇ2n�1 4 p 2 � 1 2 C1 i C � ˛2nCˇ2n 2 �� ˛2n�1Cˇ2n�1 2 � C1 9=; 8 D ˛4n.1C˛�2C2˛�1/Cˇ4n.1Cˇ�2C2ˇ�1/�4 64 D ˛4n�2.˛ˇ/2nCˇ4n 32 D � ˛2n�ˇ2n 4 p 2 �2 D Sn: The others can be proved similarly. � 3. NUMERICAL RESULTS In [8], the authors dealt with the common values of balancing–type sequences. At the end of this paper we investigate this problem between Lucas–balancing and combinatorial numbers. Let us consider the equation Cn D f .x/; (3.1) where f .x/ is a polynomial with integer coefficients and Cn is the nth Lucas– balancing number. In this paper we study (3.1) when f .x/ is one of the polynomials � x k � , and Sk.x/D 1 k C2kC�� �C .x�1/k; ˘k.x/D x.xC1/ : : : .xCk�1/ for fixed integer k � 2. We have to mention that we can deduce the following property of Lucas–balancing from equation (1.3): Lemma 7. If x D˙Cn and y D˙Bn, then the equation x2�8y2 D 1 is valid. Our numerical result is the following. BALANCING, PELL AND SQUARE TRIANGULAR FUNCTIONS 1229 Theorem 7. Let 2 � k � 4 and f .x/ be one of the polynomials � x k � , ˘k.x/, Sk�1.x/. Then the solutions .Cn;x/ of equation (3.1) are in the table Cm Bm f .x/ x k 3 1 � x k � 3 2 3 1 Sk�1.x/ 3 2 Proof. Consider the equation (3.1) when f .x/ one of polynomials � x 2 � , � x 3 � and� x 4 � . Using the multiplications by suitable powers of 2 and transformations X D x, X D 2.x�1/2C1, X D x2�3xC1 respectively, for the polynomials above. Then by Lemma 7 we obtain .23y/2 D 2X4�4X3C2X2�8;� 243y �2 DX3�7X2C15X �297;� 253y �2 D 2X4�4X2�1150; where y is a balancing number. These types of equations are solvable by MAGMA (IntegralQuarticPoints and IntegralPoints), so after testing them we get the solutions above. The suitable command at the first case is IntegralQuarticPoints([2,-4,2,0,-8],[3,8]), because the point Œ3;8� is on the curve. We get the solutions (2,8),(2,0),(1,0),(3,8) for .X;23y/. Using these results we realize that there is only one solution for Bn, Cn and x, where .Bn;Cn;x/D .1;3;3/. Let us consider the second equation. In this case there is no solution, since from the property (1.3) follows that all Lucas–balancing numbers are odd, while � x 3 � is even. We use in the third case the command IntegralQuarticPoints([2,0,-4,0,-1150]) and we found no solution. Remark 1. We mention that we can use IntegralQuarticPoints([]) in the case also when the constant term is not square, but the odd powers are missing from the right hand side. Now let f .x/ be equal to Sk�1.x/. If k D 2, then S1.x/ D � x 2 � , and its unique solutions has already been determined. Applying the transformations X D 2.2x�1/2, X D � x 2 � , respectively, to the equa- tion (3.1) when f .x/D S2.x/ and f .x/D S3.x/ we get� 263y �2 DX3�4X2C4X �4608; .22y/2 D 2X4�2: Using the commands IntegralPoints(EllipticCurve([0,-4,4,0,-4608])) and IntegralQuarticPoints([2,0,0,0,-2]), it provides only the solution .1;0/, in the second case, that is there is no solution for Lucas–balancing number. 1230 AHMET TEKCAN, MERVE TAYAT, AND PÉTER OLAJOS Finally let f .x/D˘2.x/; ˘3.x/; ˘4.x/ and by using the transformations X D 2xC1, X D 2.xC1/2C1 and X D x2C3x we arrive that .24y/2 D 2X4�4X2�30; .23y/2 DX3�7X2C15X �17; y2 D 2X4C8X3C8X2�2: By MAGMA, there exist no solution in the first and the second cases. In the third case we can easily conclude the same as follows. Since X is always even, therefore y is also even and we have y2 � 0.mod 4/ and 2X4C8X3C8X2�2� 2.mod 4/; which is impossible. � ACKNOWLEDGMENTS The authors wish to thank the anonymous referees for their detailed comments and suggestions which have improved the presentation of the paper. REFERENCES [1] A. Behera and G. K. Panda, “On the square roots of triangular numbers,” Fibonacci Q., vol. 37, no. 2, pp. 98–105, 1999. [2] K. K. Dash, R. S. Ota, and S. Dash, “t -balancing numbers,” Int. J. Contemp. Math. Sci., vol. 7, no. 41-44, pp. 1999–2012, 2012. [3] R. Finkelstein, “The house problem,” Am. Math. Mon., vol. 72, pp. 1082–1088, 1965, doi: 10.2307/2315953. [4] T. Kovács, K. Liptai, and P. Olajos, “On .a;b/-balancing numbers,” Publ. Math., vol. 77, no. 3-4, pp. 485–498, 2010. [5] K. Liptai, “Fibonacci balancing numbers,” Fibonacci Q., vol. 42, no. 4, pp. 330–340, 2004. [6] K. Liptai, “Lucas balancing numbers,” Acta Math. 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Authors’ addresses Ahmet Tekcan Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkiye E-mail address: tekcan@uludag.edu.tr Merve Tayat Uludag University, Faculty of Science, Department of Mathematics, Bursa-Turkiye E-mail address: mervetayat07@mail.com Péter Olajos University of Miskolc, Institute of Mathematics, Department of Applied Mathematics, Hungary E-mail address: matolaj@uni-miskolc.hu http://dx.doi.org/10.1155/2014/897834 http://dx.doi.org/10.5486/PMD.2013.5654 1. Introduction 2. Main results 3. Numerical results Acknowledgments References