INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS
|
|
- Nancy Newton
- 5 years ago
- Views:
Transcription
1 INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS BIJAN KUMAR PATEL, NURETTIN IRMAK and PRASANTA KUMAR RAY Communicated by Alexandru Zaharescu The aim of this article is to establish some combinatorial expressions of balancing and Lucas-balancing numbers and investigate some of their properties. AMS 010 Subect Classification: 11B37, 11B39. Key words: balancing numbers, Lucas-balancing numbers, incomplete balancing numbers, incomplete Lucas-balancing numbers. 1. INTRODUCTION The terms balancing numbers and Lucas-balancing numbers are used to describe the series of numbers generated by the recursive formulas B n = 6B n 1 B n ; B 0 = 0, B 1 = 1 with n and C n = 6C n 1 C n ; C 0 = 1, C 1 = 3, with n respectively [1,10]. The roots λ 1 = 3+ 8 and λ = 3 8 for both these sequences, the respective Binet formulas are B n = λn 1 λn λ 1 λ and C n = λ 1+λ [1, 10]. Many interesting results of balancing numbers and their related sequences can be found in [5, 10 1]. In [4], Filipponi established two interesting classes of integers namely, incomplete Fibonacci numbers and incomplete Lucas numbers which were obtained from some of the well-known combinatorial forms of Fibonacci and Lucas numbers. He has also studied some of the congruence properties for incomplete Lucas numbers in [4]. Filipponi dreamt a glimpse of possible generalizations of incomplete Fibonacci and Lucas numbers which were fulfilled by some authors later [, 7 9, 13]. In this article we establish some combinatorial expressions for balancing and Lucas-balancing numbers and introduce incomplete balancing and incomplete Lucas-balancing numbers. Balancing sequence has been generalized in many ways. One important generalization of balancing numbers is the k-balancing numbers introduced in [6, 1]. k-balancing numbers are defined recursively by B k,n+1 = 6kB k,n B k,n 1 n, MATH. REPORTS 0(70, 1 (018, 59 7
2 60 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray with initials B k,0 = 0 and B k,1 = 1. In [1], Ray has introduced sequence of balancing polynomials {B n (x} n=0that are natural extension of k-balancing numbers and is defined recursively by 1, if n = 0 B n (x = 6x, if n = 1 6xB n 1 (x B n (x, if n > 1. Further, he has also established the Binet formula as where B n (x = λn 1 (x λn (x, λ 1 (x = 3x +, λ (x = 3x and = 9x 1. The first few balancing polynomials are B (x = 6x, B 3 (x = 36x 1, B 4 (x = 16x 3 1x, B 5 (x = 196x 4 108x +1. The derivatives of balancing polynomials in the form of convolution of these polynomials are also presented in [1]. In a similar manner, the n th Lucas-balancing polynomial C n (x is defined as C n (x = and its Binet formula is 1, if n = 0 3x, if n = 1 6xC n 1 (x C n (x, if n > 1, C n (x = λn 1 (x + λn (x. The first few Lucas-balancing polynomials are C (x = 18x 1, C 3 (x = 108x 3 9x, C 4 (x = 648x 4 7x + 1, C 5 (x = 3888x 5 540x x. In this article, authors aim is to establish some combinatorial expressions of balancing and Lucas-balancing numbers and investigate some of their properties.. SOME COMBINATORIAL EXPRESSIONS OF BALANCING AND LUCAS BALANCING NUMBERS In this section, we establish some combinatorial expressions for balancing and Lucas-balancing numbers. These expressions lead to form some combinatorial expressions for balancing and Lucas-balancing polynomials.
3 3 Incomplete balancing and Lucas-balancing numbers 61 Theorem.1. Let B n and C n denote n th balancing and Lucas-balancing numbers respectively, then (.1 (. B n = (n 1/ n/ C n = 3 ( 1 where. denotes the floor function. ( 1 ( n 1 n n 6 n 1, ( n 6 n 1, Proof. From the Binet formulas for both balancing and Lucas-balancing numbers and by well-known Waring formulas [3], B n = λn 1 λn (n 1/ = λ 1 λ C n = λ n 1 + λ n = n/ ( 1 n n ( 1 ( n 1 and the result follows as λ 1 λ = 1 and λ 1 + λ = 6. (λ 1 λ (λ 1 + λ n 1, ( n (λ 1 λ (λ 1 + λ n, Indeed, the combinatorial expression for polynomial B n (x leads to B n (x = (n 1/ ( 1 ( n 1 (6x n 1 for n 1, which is shown in [1]. Similarly the combinatorial expression for polynomial C n (x will be n/ C n (x = 3 ( 1 n n ( n 6 n 1 x n for n 1. Further, their derivatives are respectively given by (n 1/ (.3 B n(x= ( 1 (n 1 n/ (.4 C n(x n(n = 3 ( 1 n ( n 1 6 n 1 x n for n 1, ( n (6x n 1 for n 1.
4 6 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray 4 Clearly, B 0 (x = C 0 (x = 0. Some simple properties of the polynomials B n(x and C n(x can be derived from the Binet formulas. In fact, letting λ 1(x = d dx (λ 1(x = 3λ 1(x, λ (x = d dx (λ (x = 3λ (x, which follows (λ n 1 (x = d dx (λn 1 (x = 3nλn 1 (x, (λn (x = d dx (λn (x = 3nλn (x, we can write B n(x = d ( λ n 1 (x λ n (x = 3nC n(x 9xB n (x (.5 dx, (.6 C n(x = d dx (λn 1 (x + λ n (x = 3nB n (x. 3. INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS The combinatorial expressions (.1 and (. give rise to two interesting classes of integers B n (k and C n (k, for integral values n and k. We call these integers as incomplete balancing numbers and incomplete Lucasbalancing numbers respectively which will be defined subsequently. In this section, authors aim to establish certain properties of incomplete balancing and incomplete Lucas-balancing numbers. Definition 3.1. The incomplete balancing numbers B n (k be defined as for any natural number n, k ( B n (k = ( 1 n 1 (3.7 6 n 1, 0 k ñ, where ñ = n 1. The numbers B n (k are shown in the Table 1 for the first few values of n and the corresponding admissible values of k. Observation of Table 1 gives rise to, We define the incomplete Lucas-balancing numbers in the following manner: B n (0 = 6 n 1 for n 1, B n (ñ = B n for n 1, { Bn ± 3n, if n is even B n (ñ 1 = B n ± 1, if n is odd for n 3.
5 5 Incomplete balancing and Lucas-balancing numbers 63 TABLE 1 Incomplete balancing numbers n/k Definition 3.. The incomplete Lucas-balancing numbers C n (k be defined as for any natural number n, k ( C n (k = 3 ( 1 n n (3.8 6 n 1, 0 k ˆn, n where ˆn = n. The numbers C n (k are shown in Table for the first few values of n and the corresponding admissible values of k. TABLE Incomplete Lucas-balancing numbers n/k Observation of Table gives rise to, C n (0 = 3.6 n 1 for n 1, C n (ˆn = C n for n 1, { Cn ± 1, if n is even C n (ˆn 1 = C n ± n, if n is odd for n 3.
6 64 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray SOME IDENTITIES CONCERNING THE INCOMPLETE BALANCING NUMBERS B n (k Like balancing and Lucas-balancing numbers, incomplete balancing numbers also satisfy the second order recurrence relation. The following result demonstrates this fact. Proposition 3.3. The numbers B n (k obey the second order recurrence relation (3.9 B n+ (k + 1 = 6B n+1 (k + 1 B n (k for 0 k (n /. Proof. By virtue of the Definition 3.7, k+1 6B n+1 (k+1 B n (k = 6 and the result follows. k+1 = 6 ( 1 ( n ( 1 [( n 6 n + k ( n 1 k+1 ( = ( 1 n n+1, ( 1 ( n 1 6 n 1 ( ] n + 6 n 6 n 1 Observe that, the relation (3.9 can be transformed into the non-homogeneous recurrence relation ( (3.10 B n+ (k = 6B n+1 (k B n (k ( 1 k+1 6 n 1 k n 1 k. k The relation (3.10 can be generalized as follows. Proposition 3.4. Let 0 k n h 1. Then the identity h ( (3.11 ( h B n+ (k + = ( 1 h+1 B n+h (k + h holds. Proof. Using induction on h, the result (3.11 clearly holds for h = 0 and h = 1. Assume that the result holds for some h > 1. For the inductive step, we have h+1 ( ( h + 1 B n+ (k +
7 7 Incomplete balancing and Lucas-balancing numbers 65 h+1 [( ( ] = ( h h + B 1 n+ (k + = ( 1 h 1 B n+h (k + h + = ( 1 h 1 B n+h (k + h + = ( 1 h 1 B n+h (k + h 6 h+1 ( ( 1 i+ 6 i+1 h B i i+n+1 (k i i= 1 h ( ( 1 i+ 6 i+1 h B i i+n+1 (k i i=0 h ( ( 1 i+1 6 i h B i i+n+1 (k i i=0 = ( 1 h 1 B n+h (k + h 6( 1 h 1 B n+h+1 (k h = ( 1 h 1 [B n+h (k + h 6B n+h+1 (k h] = ( 1 h B n+(h+1 (k h, which completes the proof. Using induction on m, the following identity can be proved analogously. h m ( ( m h (3.1 B m + n+ ( = ( 1 h+1 B n+h m (h m, where h m and n h m + 1. Notice that, the identity obtained from (3.1 by setting m = 0 is identical to the identity obtained from (3.11 for k = 0. The following is an interesting relation concerning incomplete balancing numbers. The sum of all elements of the n th row of the array of Table 1 is expressed in terms of balancing and Lucas-balancing numbers. Proposition 3.5. For incomplete balancing numbers B n (k, the following identity holds. ñ { (3nCn B B n (k = n /16, if n is even (3nC n + 7B n /16 if n is odd. k=0 Proof. Recall that ñ = n 1. Then, B n (0 + B n (1 + + B n (ñ ( n 1 0 = 6 n [ ( ( 1 0 n [ ( ( 1 0 n n ( 1ñ 0 ( 6 n 1 0 +( 1 1 n ( n 1 ñ 6 n 1 ñ] ñ 6 n 1 ]
8 66 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray 8 ( n 1 0 = (ñ ( n 1 ñ + + ( 1ñ ñ ñ ( = (ñ + 1 ( 1 n 1 = (ñ + 1 ñ = (ñ + 1B n ( 6 n (ñ + 1 1( 1 1 n ( 1 ( n 1 6 n 1 ñ 6 n 1 6 n 1 ñ ( ( 1 n 1 6 n 1. 6 n 1 ñ ( ( 1 n 1 6 n 1 The second term of the right hand side expression of the above equation ñ leads to Bn(8n+1 3nCn 16 for x = 1 in (.3 and (.5. For n is even, B n (k is nb n Bn(8n+1 3nCn 16 and for n is odd, This completes the proof. ñ k=0 k=0 B n (k is (n+1bn Bn(8n+1 3nCn 16. In [4], Filipponi has shown that for n = m, where m is a non-negative integer, the incomplete Fibonacci numbers F n (k is odd for all admissible values of k. The following result shows that for n = m, the incomplete balancing numbers B n (k is even for all admissible values of k. Proposition 3.6. If n = m for any non-negative integer m, then B n (k is even for all admissible values of k. Proof. We prove this result by induction on m. The basic step is true for m = 1. In the inductive step, assume that the result is valid for all m n. Now, using the (3.9; we have B m+ (k + 1 = 6B m+1 (k + 1 B m (k. Clearly the first term is divisible by and by the hypothesis the second term is also even and hence the result follows. Filipponi has also shown that for any prime p, the incomplete Lucas numbers satisfy the identity L p (k 1 (mod p for all admissible values of k in [4]. Whereas no such type of identity exists in incomplete Fibonacci numbers. To demonstrate this fact, consider the following example.
9 9 Incomplete balancing and Lucas-balancing numbers 67 Example 3.7. Let p = 7. Consider k = 1,, 3 and observe that F 7 (1 = 6 1 (mod 7, F 7 ( = 1 (mod 7 and F 7 (3 = 13 1 (mod 7, respectively. However, the similar type of identities do exist in both incomplete balancing and Lucas-balancing numbers. The following result demonstrates this fact. Proposition 3.8. If n = m for any non-negative integer m, then for all admissible values of k, B n (k 0 (mod n. Proof. By virtue of Definition (3.7, (3.13 B n (k = 6 m 1 + k =1 ( 1 ( m 1 6 m 1. The first term on the right hand side (3.13 is an integer when B m(k is divisible by m for all m 0. The second term is also an integer due to Filipponi [3], and thus the congruence follows. 3.. SOME IDENTITIES INVOLVING THE INCOMPLETE LUCAS-BALANCING NUMBERS C n (k Balancing numbers and Lucas-balancing numbers are related by an identity B n+1 B n 1 = C n. Similar properties are valid for incomplete balancing and Lucas-balancing numbers. Proposition 3.9. The identity (3.14 B n+1 (k B n 1 (k 1 = C n (k, holds for 0 k ˆn, where ˆn = n. Proof. By virtue of Definition 3.7, k ( B n+1 (k B n 1 (k 1 = ( 1 n k 1 6 n = = k k ( 1 ( n ( 1 ( n n + k =1 ( 1 ( n ( 1 1 ( n 1 1 ( n 6 n 6 n 6 n
10 68 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray 10 which ends the proof. = 3 k ( 1 n n ( n n 1 Proposition The numbers C n (k obey the second order recurrence relation (3.15 C n+ (k + 1 = 6C n+1 (k + 1 C n (k. Proof. Using (3.14 and (3.9, C n+ (k + 1 = 1 {B n+3(k + 1 B n+1 (k} and the result follows. = 1 [{6B n+(k + 1 B n+1 (k} {6B n (k B n 1 (k 1}] = 3(B n+ (k + 1 B n (k + 1 (B n 1(k 1 B n+1 (k = 6C n+1 (k + 1 C n (k, Notice that, the relation (3.15 can be transformed into the non-homogeneous recurrence relation (3.16 ( C n+ (k = 6C n+1 (k C n (k + ( 1 k+ 3n n k 6 n k 1 for n k. n k k Proposition For 0 k ñ, then the following identity holds. ( C n (k = B n+ (k B n (k. Proof. In view of (3.14 and (3.15, from which the result follows. B n+ (k = C n+1 (k + B n (k 1, B n (k = B n (k 1 C n 1 (k 1, Using the relations (3.11 and (3.14, the identity (3.15 can be generalized as follows. holds. Proposition 3.1. For 0 k n h 1, then h ( ( h C n+ (k + = ( 1 h+1 C n+h (k + h,
11 11 Incomplete balancing and Lucas-balancing numbers 69 The following is an interesting relation concerning incomplete Lucasbalancing numbers. The sum of all elements of the n th row of the array of Table is expressed in terms of balancing and Lucas-balancing numbers. Proposition For incomplete Lucas-balancing numbers C n (k, ˆn k=0 C n (k = { Cn + n(3b n C n /, if n is even {C n + n(3b n C n }/, if n is odd. Proof. Recall that ˆn = n. Therefore, we have C n (0 + C n (1 + + C n (ˆn = (ˆn + 1C n 3 ˆn ( 1 n n ( n 6 n 1. Putting x = 1 in (.4 and (.6 and adopting the same procedure described earlier, we obtain B n = ˆn ( 1 n n ( n 6 n 1. Therefore, 3 ˆn ( 1 n n ( n 6 n 1 = 3n(C n B n /. For n is even and odd, ˆn k=0 C n (k is ( n + 1C n 3n(C n B n / and ( n+1 C n 3n(C n B n / respectively. Therefore, the proof is completed. Proposition If n = 3 m for any non-negative integer m, then for all admissible values of k, C n (k 0 (mod n. Proof. By virtue of Definition 3.16, k ( (3.18 C n (k = 3m 1 3 3m + 3 m+1 ( m 3 m 6 3m 1. =1 The first term on the right hand side of (3.18 is an integer when C 3 m(k is divisible by 3 m for all m 0. The second term is also an integer again due to Filipponi [4] and thus the congruence follows.
12 70 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray 1 4. GENERATING FUNCTIONS OF THE INCOMPLETE BALANCING AND LUCAS-BALANCING NUMBERS The generating functions for balancing and Lucas-balancing numbers play a vital role to find out many important identities for these numbers. In this section, we derive generating functions for both incomplete balancing and Lucas-balancing numbers. The following result [7] is useful to prove the subsequent theorem. Lemma 4.1. Let {s n } n N be a complex sequence satisfying the following non-homogeneous second-order recurrence relation: s n = as n 1 + bs n + r n, n > 1, where a, b C and r n : N C is a given sequence. Then the generating function U(t of s n is U(t = G(t + s 0 r 0 + (s 1 s 0 a r 1 t 1 at bt, where G(t denotes generating function of r n. The following are the generating functions for incomplete balancing and incomplete Lucas-balancing numbers. Theorem 4.. Let k be a fixed positive integer. Then B k (t = t k+1 (1 6tk+1 (B k+1 B k t ( 1 k+1 t (4.19 (1 6t + t (1 6t k+1, (4.0 C k (t = t k (1 6tk+1 (C k C k 1 t ( 1 k+1 t (1 6t + t (1 6t k+1. Proof. By virtue of the identity (3.7, for 0 n < k + 1, B n (k = 0 and for other values of n and r 0 integers, then B k+1+r (k = B k+r. It follows from (3.10 that, for n k + 3, B n (k = 6B n 1 (k B n (k ( 1 k+1 ( n 3 k n 3 k 6 n 3 k. Letting s 0 = B k+1 (k, s 1 = B k+ (k, (..., s n = B n+k+1 and suppose that n + k r 0 = r 1 = 0 and r n = ( 1 k+1 6 n. It is easy to deduce the n generating function of the sequence {r n } as G(t = ( 1k+1 t. Therefore by (1 6t k+1 Lemma 4.1, the generating function of the incomplete balancing numbers satisfies the following: U k (t(1 6t + t + ( 1k+1 t (1 6t k+1 = B k+1 + (B k+ 6B k+1 t,
13 13 Incomplete balancing and Lucas-balancing numbers 71 where U k (t is the generating function of the sequence {s n }. It follows that B k (t = t k+1 U k (t. In the proof of the identity (4.0, we use the following facts: For 0 n < k, C n (k = 0, and for other values of n and r 0 integers then, C k+1+r (k = C k+r. It follows from (3.16 that, for n k +, C n (k = 6C n 1 (k C n (k + ( 1 k+ 3(n (n k ( n k n k 6 n k 3. Letting s 0 = C k (k, s 1 = C k+1 (k,..., s n = C n+k and suppose that r 0 = r 1 = 0 and ( r n = 3.6 n 3 k+ n + k n + k ( 1. n + k n The generating function of the sequence {r n } is G 1 (t = ( 1k+ 3t ( t. Again (1 6t k+1 from Lemma 4.1, the generating function of the incomplete Lucas-balancing numbers satisfies the equation: U k (t(1 6t + t + ( 1k+ 3t ( t (1 6t k+1 = C k + (C k+1 6C k t. Finally, we conclude that C k (t = t k U k (t. Acknowledgments. The authors would like to express their gratitude to the anonymous referee for his valuable comments and suggestions which improved the presentation of the article to a great extent. REFERENCES [1] A. Behera and G.K. Panda, On the square roots of triangular numbers. Fibonacci Quart. 37 (1999,, [] G.B. Dordevic and H.M. Srivastava, Incomplete generalized Jacobsthal and Jacobsthal- Lucas numbers. Math. Comput. Modelling 4 (005, [3] P. Filipponi, Waring s formula, the binomial formula, and generalized Fibonacci matrices. Fibonacci Quart. 30 (199, 3, [4] P. Filipponi, Incomplete Fibonacci and Lucas numbers. Rend. Circ. Mat. Palermo. 45 (1996,, [5] R. Keskin and O. Karaath, Some new properties of balancing numbers and square triangular numbers. J. Integer Seq. 15 (01, Article [6] A. Özkoc, Tridigonal Matrices via k-balancing Number. British J. of Math. Comput. Sci. 10 (015, 4, [7] Á. Pinter and H.M. Srivastava, Generating functions of the incomplete Fibonacci and Lucas numbers. Rend. Circ. Mat. Palermo ( 48 (1999,,
14 7 Bian Kumar Patel, Nurettin Irmak and Prasanta Kumar Ray 14 [8] J. Ramirez, Bi-periodic incomplete Fibonacci sequence. Ann. Math. Inform. 4 (013, [9] J. Ramirez, Incomplete generalized Fibonacci and Lucas polynomials. arxiv: , 013. [10] P.K. Ray, Balancing and cobalancing numbers. Ph.D. thesis, Department of Mathematics, National Institute of Technology, Rourkela, India, 009. [11] P.K. Ray, Some congruences for balancing and Lucas balancing numbers and their applications. Integers 14 (014, #A8. [1] P.K. Ray, On the properties of k-balancing numbers. Ain Shams Eng. J., 016. In press. [13] D. Tasci and M. Cetin Firengiz, Incomplete Fibonacci and Lucas p-numbers. Math. Comput. Modelling 5(9-10 (010, Received 3 February 016 IIIT Bhubaneswar, Department of Mathematics, Bhubaneswar iiit.bian@gmail.com Niǧde University, Department of Mathematics, Niǧde 5140, Turkey nirmak@nigde.edu.tr Sambalpur University, School of Mathematical Sciences, Sambalpur , India prasantamath@suniv.ac.in
RECIPROCAL SUMS OF SEQUENCES INVOLVING BALANCING AND LUCAS-BALANCING NUMBERS
RECIPROCAL SUMS OF SEQUENCES INVOLVING BALANCING AND LUCAS-BALANCING NUMBERS GOPAL KRISHNA PANDA, TAKAO KOMATSU and RAVI KUMAR DAVALA Communicated by Alexandru Zaharescu Many authors studied bounds for
More informationCertain Diophantine equations involving balancing and Lucas-balancing numbers
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 0, Number, December 016 Available online at http://acutm.math.ut.ee Certain Diophantine equations involving balancing and Lucas-balancing
More informationIncomplete Tribonacci Numbers and Polynomials
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 17 (014, Article 14.4. Incomplete Tribonacci Numbers and Polynomials José L. Ramírez 1 Instituto de Matemáticas y sus Aplicaciones Calle 74 No. 14-14 Bogotá
More informationOn arithmetic functions of balancing and Lucas-balancing numbers
MATHEMATICAL COMMUNICATIONS 77 Math. Commun. 24(2019), 77 1 On arithmetic functions of balancing and Lucas-balancing numbers Utkal Keshari Dutta and Prasanta Kumar Ray Department of Mathematics, Sambalpur
More informationNEW IDENTITIES FOR THE COMMON FACTORS OF BALANCING AND LUCAS-BALANCING NUMBERS
International Journal of Pure and Applied Mathematics Volume 85 No. 3 013, 487-494 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.173/ijpam.v85i3.5
More informationBalancing sequences of matrices with application to algebra of balancing numbers
Notes on Number Theory and Discrete Mathematics ISSN 1310 5132 Vol 20 2014 No 1 49 58 Balancing sequences of matrices with application to algebra of balancing numbers Prasanta Kumar Ray International Institute
More informationA trigonometry approach to balancing numbers and their related sequences. Prasanta Kumar Ray 1
ISSN: 2317-0840 A trigonometry approach to balancing numbers and their related sequences Prasanta Kumar Ray 1 1 Veer Surendra Sai University of Technology Burla India Abstract: The balancing numbers satisfy
More informationOn repdigits as product of consecutive Lucas numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 5 102 DOI: 10.7546/nntdm.2018.24.3.5-102 On repdigits as product of consecutive Lucas numbers
More informationBalancing And Lucas-balancing Numbers With Real Indices
Balancing And Lucas-balancing Numbers With Real Indices A thesis submitted by SEPHALI TANTY Roll No. 413MA2076 for the partial fulfilment for the award of the degree Master Of Science Under the supervision
More informationPELLANS SEQUENCE AND ITS DIOPHANTINE TRIPLES. Nurettin Irmak and Murat Alp
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 100114 016, 59 69 DOI: 10.98/PIM161459I PELLANS SEQUENCE AND ITS DIOPHANTINE TRIPLES Nurettin Irmak and Murat Alp Abstract. We introduce a novel
More informationP. K. Ray, K. Parida GENERALIZATION OF CASSINI FORMULAS FOR BALANCING AND LUCAS-BALANCING NUMBERS
Математичнi Студiї. Т.42, 1 Matematychni Studii. V.42, No.1 УДК 511.217 P. K. Ray, K. Parida GENERALIZATION OF CASSINI FORMULAS FOR BALANCING AND LUCAS-BALANCING NUMBERS P. K. Ray, K. Parida. Generalization
More informationq-counting hypercubes in Lucas cubes
Turkish Journal of Mathematics http:// journals. tubitak. gov. tr/ math/ Research Article Turk J Math (2018) 42: 190 203 c TÜBİTAK doi:10.3906/mat-1605-2 q-counting hypercubes in Lucas cubes Elif SAYGI
More informationARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY OF THE DIOPHANTINE EQUATION 8x = y 2
International Conference in Number Theory and Applications 01 Department of Mathematics, Faculty of Science, Kasetsart University Speaker: G. K. Panda 1 ARITHMETIC PROGRESSION OF SQUARES AND SOLVABILITY
More informationApplication of Balancing Numbers in Effectively Solving Generalized Pell s Equation
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 2, Issue 2, February 2014, PP 156-164 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org Application
More informationBALANCING NUMBERS : SOME IDENTITIES KABERI PARIDA. Master of Science in Mathematics. Dr. GOPAL KRISHNA PANDA
BALANCING NUMBERS : SOME IDENTITIES A report submitted by KABERI PARIDA Roll No: 1MA073 for the partial fulfilment for the award of the degree of Master of Science in Mathematics under the supervision
More informationMATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences.
MATH 433 Applied Algebra Lecture 4: Modular arithmetic (continued). Linear congruences. Congruences Let n be a postive integer. The integers a and b are called congruent modulo n if they have the same
More informationON GENERALIZED BALANCING SEQUENCES
ON GENERALIZED BALANCING SEQUENCES ATTILA BÉRCZES, KÁLMÁN LIPTAI, AND ISTVÁN PINK Abstract. Let R i = R(A, B, R 0, R 1 ) be a second order linear recurrence sequence. In the present paper we prove that
More informationLinks Between Sums Over Paths in Bernoulli s Triangles and the Fibonacci Numbers
Links Between Sums Over Paths in Bernoulli s Triangles and the Fibonacci Numbers arxiv:1611.09181v1 [math.co] 28 Nov 2016 Denis Neiter and Amsha Proag Ecole Polytechnique Route de Saclay 91128 Palaiseau
More informationGENERATING FUNCTIONS K-FIBONACCI AND K-JACOBSTHAL NUMBERS AT NEGATIVE INDICES
Electronic Journal of Mathematical Analysis and Applications Vol. 6(2) July 2018, pp. 195-202. ISSN: 2090-729X(online) http://fcag-egypt.com/journals/ejmaa/ GENERATING FUNCTIONS K-FIBONACCI AND K-JACOBSTHAL
More informationA Recursive Relation for Weighted Motzkin Sequences
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 8 (005), Article 05.1.6 A Recursive Relation for Weighted Motzkin Sequences Wen-jin Woan Department of Mathematics Howard University Washington, D.C. 0059
More informationDivisibility in the Fibonacci Numbers. Stefan Erickson Colorado College January 27, 2006
Divisibility in the Fibonacci Numbers Stefan Erickson Colorado College January 27, 2006 Fibonacci Numbers F n+2 = F n+1 + F n n 1 2 3 4 6 7 8 9 10 11 12 F n 1 1 2 3 8 13 21 34 89 144 n 13 14 1 16 17 18
More informationLegendre s Equation. PHYS Southern Illinois University. October 18, 2016
Legendre s Equation PHYS 500 - Southern Illinois University October 18, 2016 PHYS 500 - Southern Illinois University Legendre s Equation October 18, 2016 1 / 11 Legendre s Equation Recall We are trying
More informationON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS
Hacettepe Journal of Mathematics and Statistics Volume 8() (009), 65 75 ON QUADRAPELL NUMBERS AND QUADRAPELL POLYNOMIALS Dursun Tascı Received 09:0 :009 : Accepted 04 :05 :009 Abstract In this paper we
More informationCullen Numbers in Binary Recurrent Sequences
Cullen Numbers in Binary Recurrent Sequences Florian Luca 1 and Pantelimon Stănică 2 1 IMATE-UNAM, Ap. Postal 61-3 (Xangari), CP 58 089 Morelia, Michoacán, Mexico; e-mail: fluca@matmor.unam.mx 2 Auburn
More informationPartition of Integers into Distinct Summands with Upper Bounds. Partition of Integers into Even Summands. An Example
Partition of Integers into Even Summands We ask for the number of partitions of m Z + into positive even integers The desired number is the coefficient of x m in + x + x 4 + ) + x 4 + x 8 + ) + x 6 + x
More informationOn identities with multinomial coefficients for Fibonacci-Narayana sequence
Annales Mathematicae et Informaticae 49 08 pp 75 84 doi: 009/ami080900 http://amiuni-eszterhazyhu On identities with multinomial coefficients for Fibonacci-Narayana sequence Taras Goy Vasyl Stefany Precarpathian
More informationRICCATI MEETS FIBONACCI
Wolfdieter Lang Institut für Theoretische Physik, Universität Karlsruhe Kaiserstrasse 2, D-763 Karlsruhe, Germany e-mail: wolfdieter.lang@physik.uni-karlsruhe.de (Submitted November 200- Final Revision
More informationSome Congruences for the Partial Bell Polynomials
3 47 6 3 Journal of Integer Seuences, Vol. 009), Article 09.4. Some Congruences for the Partial Bell Polynomials Miloud Mihoubi University of Science and Technology Houari Boumediene Faculty of Mathematics
More informationTHE p-adic VALUATION OF LUCAS SEQUENCES
THE p-adic VALUATION OF LUCAS SEQUENCES CARLO SANNA Abstract. Let (u n) n 0 be a nondegenerate Lucas sequence with characteristic polynomial X 2 ax b, for some relatively prime integers a and b. For each
More informationIn memoriam Péter Kiss
Kálmán Liptai Eszterházy Károly Collage, Leányka út 4, 3300 Eger, Hungary e-mail: liptaik@gemini.ektf.hu (Submitted March 2002-Final Revision October 2003) In memoriam Péter Kiss ABSTRACT A positive integer
More informationProblem Set 5 Solutions
Problem Set 5 Solutions Section 4.. Use mathematical induction to prove each of the following: a) For each natural number n with n, n > + n. Let P n) be the statement n > + n. The base case, P ), is true
More informationOn New Identities For Mersenne Numbers
Applied Mathematics E-Notes, 18018), 100-105 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ On New Identities For Mersenne Numbers Taras Goy Received April 017 Abstract
More informationMULTI-RESTRAINED STIRLING NUMBERS
MULTI-RESTRAINED STIRLING NUMBERS JI YOUNG CHOI DEPARTMENT OF MATHEMATICS SHIPPENSBURG UNIVERSITY SHIPPENSBURG, PA 17257, U.S.A. Abstract. Given positive integers n, k, and m, the (n, k)-th m- restrained
More informationCALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p
CALCULATING EXACT CYCLE LENGTHS IN THE GENERALIZED FIBONACCI SEQUENCE MODULO p DOMINIC VELLA AND ALFRED VELLA. Introduction The cycles that occur in the Fibonacci sequence {F n } n=0 when it is reduced
More informationABSTRACT 1. INTRODUCTION
THE FIBONACCI NUMBER OF GENERALIZED PETERSEN GRAPHS Stephan G. Wagner Department of Mathematics, Graz University of Technology, Steyrergasse 30, A-8010 Graz, Austria e-mail: wagner@finanz.math.tu-graz.ac.at
More informationCONGRUENCES FOR BERNOULLI - LUCAS SUMS
CONGRUENCES FOR BERNOULLI - LUCAS SUMS PAUL THOMAS YOUNG Abstract. We give strong congruences for sums of the form N BnVn+1 where Bn denotes the Bernoulli number and V n denotes a Lucas sequence of the
More informationON EXPLICIT FORMULAE AND LINEAR RECURRENT SEQUENCES. 1. Introduction
ON EXPLICIT FORMULAE AND LINEAR RECURRENT SEQUENCES R. EULER and L. H. GALLARDO Abstract. We notice that some recent explicit results about linear recurrent sequences over a ring R with 1 were already
More informationCATALAN TRIANGLE NUMBERS AND BINOMIAL COEFFICIENTS arxiv:6006685v2 [mathco] 7 Oct 207 KYU-HWAN LEE AND SE-JIN OH Abstract The binomial coefficients and Catalan triangle numbers appear as weight multiplicities
More informationImpulse Response Sequences and Construction of Number Sequence Identities
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 16 (013), Article 13.8. Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan
More informationCOMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS
LE MATEMATICHE Vol LXXIII 2018 Fasc I, pp 179 189 doi: 104418/201873113 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS MURAT SAHIN - ELIF TAN - SEMIH YILMAZ Let {a i },{b i } be real numbers for 0 i r 1,
More informationDISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k
DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3 k RALF BUNDSCHUH AND PETER BUNDSCHUH Dedicated to Peter Shiue on the occasion of his 70th birthday Abstract. Let F 0 = 0,F 1 = 1, and F n = F n 1 +F
More informationON A FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-LUCAS SEQUENCE. A. A. Wani, V. H. Badshah, S. Halici, P. Catarino
Acta Universitatis Apulensis ISSN: 158-539 http://www.uab.ro/auajournal/ No. 53/018 pp. 41-54 doi: 10.17114/j.aua.018.53.04 ON A FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-LUCAS SEQUENCE A. A. Wani, V.
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationF. T. HOWARD AND CURTIS COOPER
SOME IDENTITIES FOR r-fibonacci NUMBERS F. T. HOWARD AND CURTIS COOPER Abstract. Let r 1 be an integer. The r-generalized Fibonacci sequence {G n} is defined as 0, if 0 n < r 1; G n = 1, if n = r 1; G
More information9 MODULARITY AND GCD PROPERTIES OF GENERALIZED FIBONACCI NUMBERS
#A55 INTEGERS 14 (2014) 9 MODULARITY AND GCD PROPERTIES OF GENERALIZED FIBONACCI NUMBERS Rigoberto Flórez 1 Department of Mathematics and Computer Science, The Citadel, Charleston, South Carolina rigo.florez@citadel.edu
More informationG. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES
Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 3 (2007) G. Sburlati GENERALIZED FIBONACCI SEQUENCES AND LINEAR RECURRENCES Abstract. We analyze the existing relations among particular classes of generalized
More informationThe k-fibonacci matrix and the Pascal matrix
Cent Eur J Math 9(6 0 403-40 DOI: 0478/s533-0-0089-9 Central European Journal of Mathematics The -Fibonacci matrix and the Pascal matrix Research Article Sergio Falcon Department of Mathematics and Institute
More informationOn Some Identities and Generating Functions
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 38, 1877-1884 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.35131 On Some Identities and Generating Functions for k- Pell Numbers Paula
More informationMathematical Induction
Mathematical Induction MAT30 Discrete Mathematics Fall 018 MAT30 (Discrete Math) Mathematical Induction Fall 018 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)
More informationarxiv: v1 [math.co] 20 Aug 2015
arxiv:1508.04953v1 [math.co] 20 Aug 2015 On Polynomial Identities for Recursive Sequences Ivica Martinak and Iva Vrsalko Faculty of Science University of Zagreb Bienička cesta 32, HR-10000 Zagreb Croatia
More information1. Introduction Definition 1.1. Let r 1 be an integer. The r-generalized Fibonacci sequence {G n } is defined as
SOME IDENTITIES FOR r-fibonacci NUMBERS F. T. HOWARD AND CURTIS COOPER Abstract. Let r 1 be an integer. The r-generalized Fibonacci sequence {G n} is defined as 8 >< 0, if 0 n < r 1; G n = 1, if n = r
More informationSeries of Error Terms for Rational Approximations of Irrational Numbers
2 3 47 6 23 Journal of Integer Sequences, Vol. 4 20, Article..4 Series of Error Terms for Rational Approximations of Irrational Numbers Carsten Elsner Fachhochschule für die Wirtschaft Hannover Freundallee
More informationDecomposition of Pascal s Kernels Mod p s
San Jose State University SJSU ScholarWorks Faculty Publications Mathematics January 2002 Decomposition of Pascal s Kernels Mod p s Richard P. Kubelka San Jose State University, richard.kubelka@ssu.edu
More informationOn the Pell Polynomials
Applied Mathematical Sciences, Vol. 5, 2011, no. 37, 1833-1838 On the Pell Polynomials Serpil Halici Sakarya University Department of Mathematics Faculty of Arts and Sciences 54187, Sakarya, Turkey shalici@sakarya.edu.tr
More informationGeneralized Bivariate Lucas p-polynomials and Hessenberg Matrices
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.4 Generalized Bivariate Lucas p-polynomials and Hessenberg Matrices Kenan Kaygisiz and Adem Şahin Department of Mathematics Faculty
More informationEven perfect numbers among generalized Fibonacci sequences
Even perfect numbers among generalized Fibonacci sequences Diego Marques 1, Vinicius Vieira 2 Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil Abstract A positive integer
More informationM381 Number Theory 2004 Page 1
M81 Number Theory 2004 Page 1 [[ Comments are written like this. Please send me (dave@wildd.freeserve.co.uk) details of any errors you find or suggestions for improvements. ]] Question 1 20 = 2 * 10 +
More informationarxiv: v2 [math.nt] 29 Jul 2017
Fibonacci and Lucas Numbers Associated with Brocard-Ramanujan Equation arxiv:1509.07898v2 [math.nt] 29 Jul 2017 Prapanpong Pongsriiam Department of Mathematics, Faculty of Science Silpakorn University
More informationTHE GENERALIZED TRIBONACCI NUMBERS WITH NEGATIVE SUBSCRIPTS
#A3 INTEGERS 14 (014) THE GENERALIZED TRIBONACCI NUMBERS WITH NEGATIVE SUBSCRIPTS Kantaphon Kuhapatanakul 1 Dept. of Mathematics, Faculty of Science, Kasetsart University, Bangkok, Thailand fscikpkk@ku.ac.th
More informationGENERALIZATION OF AN IDENTITY OF ANDREWS
GENERALIZATION OF AN IDENTITY OF ANDREWS arxiv:math/060480v1 [math.co 1 Apr 006 Eduardo H. M. Brietze Instituto de Matemática UFRGS Caixa Postal 15080 91509 900 Porto Alegre, RS, Brazil email: brietze@mat.ufrgs.br
More informationMcGill University Faculty of Science. Solutions to Practice Final Examination Math 240 Discrete Structures 1. Time: 3 hours Marked out of 60
McGill University Faculty of Science Solutions to Practice Final Examination Math 40 Discrete Structures Time: hours Marked out of 60 Question. [6] Prove that the statement (p q) (q r) (p r) is a contradiction
More informationNotes on Systems of Linear Congruences
MATH 324 Summer 2012 Elementary Number Theory Notes on Systems of Linear Congruences In this note we will discuss systems of linear congruences where the moduli are all different. Definition. Given the
More informationOn Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions
Applied Mathematical Sciences, Vol. 9, 015, no. 5, 595-607 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.5163 On Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions
More informationCatalan triangle numbers and binomial coefficients
Contemporary Mathematics Volume 73, 208 https://doiorg/0090/conm/73/435 Catalan triangle numbers and binomial coefficients Kyu-Hwan Lee and Se-jin Oh Abstract The binomial coefficients and Catalan triangle
More informationc 2010 Melissa Ann Dennison
c 2010 Melissa Ann Dennison A SEQUENCE RELATED TO THE STERN SEQUENCE BY MELISSA ANN DENNISON DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in
More informationSOME IDENTITIES INVOLVING THE FIBONACCI POLYNOMIALS*
* Yi Yuan and Wenpeeg Zhang Research Center for Basic Science, Xi'an Jiaotong University, Xi'an Shaanxi. P.R. China (Submitted June 2000-Final Revision November 2000) 1. INTROBUCTION AND RESULTS As usual,
More informationGENERALIZED LUCAS NUMBERS OF THE FORM 5kx 2 AND 7kx 2
Bull. Korean Math. Soc. 52 (2015), No. 5, pp. 1467 1480 http://dx.doi.org/10.4134/bkms.2015.52.5.1467 GENERALIZED LUCAS NUMBERS OF THE FORM 5kx 2 AND 7kx 2 Olcay Karaatlı and Ref ik Kesk in Abstract. Generalized
More informationEVERY NATURAL NUMBER IS THE SUM OF FORTY-NINE PALINDROMES
#A3 INTEGERS 16 (2016) EVERY NATURAL NUMBER IS THE SUM OF FORTY-NINE PALINDROMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri bankswd@missouri.edu Received: 9/4/15,
More informationINDUCTION AND RECURSION. Lecture 7 - Ch. 4
INDUCTION AND RECURSION Lecture 7 - Ch. 4 4. Introduction Any mathematical statements assert that a property is true for all positive integers Examples: for every positive integer n: n!
More information#A5 INTEGERS 17 (2017) THE 2-ADIC ORDER OF SOME GENERALIZED FIBONACCI NUMBERS
#A5 INTEGERS 7 (207) THE 2-ADIC ORDER OF SOME GENERALIZED FIBONACCI NUMBERS Tamás Lengyel Mathematics Department, Occidental College, Los Angeles, California lengyel@oxy.edu Diego Marques Departamento
More informationOn Generalized Fibonacci Polynomials and Bernoulli Numbers 1
1 3 47 6 3 11 Journal of Integer Sequences, Vol 8 005), Article 0553 On Generalized Fibonacci Polynomials and Bernoulli Numbers 1 Tianping Zhang Department of Mathematics Northwest University Xi an, Shaanxi
More informationOutline. We will cover (over the next few weeks) Induction Strong Induction Constructive Induction Structural Induction
Outline We will cover (over the next few weeks) Induction Strong Induction Constructive Induction Structural Induction Induction P(1) ( n 2)[P(n 1) P(n)] ( n 1)[P(n)] Why Does This Work? I P(1) ( n 2)[P(n
More informationDiscrete Math, Spring Solutions to Problems V
Discrete Math, Spring 202 - Solutions to Problems V Suppose we have statements P, P 2, P 3,, one for each natural number In other words, we have the collection or set of statements {P n n N} a Suppose
More informationTakao Komatsu School of Mathematics and Statistics, Wuhan University
Degenerate Bernoulli polynomials and poly-cauchy polynomials Takao Komatsu School of Mathematics and Statistics, Wuhan University 1 Introduction Carlitz [6, 7] defined the degenerate Bernoulli polynomials
More information#A87 INTEGERS 18 (2018) A NOTE ON FIBONACCI NUMBERS OF EVEN INDEX
#A87 INTEGERS 8 (208) A NOTE ON FIBONACCI NUMBERS OF EVEN INDEX Achille Frigeri Dipartimento di Matematica, Politecnico di Milano, Milan, Italy achille.frigeri@polimi.it Received: 3/2/8, Accepted: 0/8/8,
More information#A91 INTEGERS 18 (2018) A GENERALIZED BINET FORMULA THAT COUNTS THE TILINGS OF A (2 N)-BOARD
#A91 INTEGERS 18 (2018) A GENERALIZED BINET FORMULA THAT COUNTS THE TILINGS OF A (2 N)-BOARD Reza Kahkeshani 1 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan,
More informationON SUMS AND RECIPROCAL SUM OF GENERALIZED FIBONACCI NUMBERS BISHNU PADA MANDAL. Master of Science in Mathematics
ON SUMS AND RECIPROCAL SUM OF GENERALIZED FIBONACCI NUMBERS A report submitted by BISHNU PADA MANDAL Roll No: 42MA2069 for the partial fulfilment for the award of the degree of Master of Science in Mathematics
More informationInduction. Induction. Induction. Induction. Induction. Induction 2/22/2018
The principle of mathematical induction is a useful tool for proving that a certain predicate is true for all natural numbers. It cannot be used to discover theorems, but only to prove them. If we have
More information#A66 INTEGERS 14 (2014) SMITH NUMBERS WITH EXTRA DIGITAL FEATURES
#A66 INTEGERS 14 (2014) SMITH NUMBERS WITH EXTRA DIGITAL FEATURES Amin Witno Department of Basic Sciences, Philadelphia University, Jordan awitno@gmail.com Received: 12/7/13, Accepted: 11/14/14, Published:
More informationBinomial coefficients and k-regular sequences
Binomial coefficients and k-regular sequences Eric Rowland Hofstra University New York Combinatorics Seminar CUNY Graduate Center, 2017 12 22 Eric Rowland Binomial coefficients and k-regular sequences
More informationMath 192r, Problem Set #3: Solutions
Math 192r Problem Set #3: Solutions 1. Let F n be the nth Fibonacci number as Wilf indexes them (with F 0 F 1 1 F 2 2 etc.). Give a simple homogeneous linear recurrence relation satisfied by the sequence
More informationCounting Palindromic Binary Strings Without r-runs of Ones
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 16 (013), Article 13.8.7 Counting Palindromic Binary Strings Without r-runs of Ones M. A. Nyblom School of Mathematics and Geospatial Science RMIT University
More informationFall 2017 Test II review problems
Fall 2017 Test II review problems Dr. Holmes October 18, 2017 This is a quite miscellaneous grab bag of relevant problems from old tests. Some are certainly repeated. 1. Give the complete addition and
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 4 1 Principle of Mathematical Induction 2 Example 3 Base Case 4 Inductive Hypothesis 5 Inductive Step When Induction Isn t Enough
More informationChapter 5.1: Induction
Chapter.1: Induction Monday, July 1 Fermat s Little Theorem Evaluate the following: 1. 1 (mod ) 1 ( ) 1 1 (mod ). (mod 7) ( ) 8 ) 1 8 1 (mod ). 77 (mod 19). 18 (mod 1) 77 ( 18 ) 1 1 (mod 19) 18 1 (mod
More information(e) Commutativity: a b = b a. (f) Distributivity of times over plus: a (b + c) = a b + a c and (b + c) a = b a + c a.
Math 299 Midterm 2 Review Nov 4, 2013 Midterm Exam 2: Thu Nov 7, in Recitation class 5:00 6:20pm, Wells A-201. Topics 1. Methods of proof (can be combined) (a) Direct proof (b) Proof by cases (c) Proof
More information#A6 INTEGERS 17 (2017) AN IMPLICIT ZECKENDORF REPRESENTATION
#A6 INTEGERS 17 (017) AN IMPLICIT ZECKENDORF REPRESENTATION Martin Gri ths Dept. of Mathematical Sciences, University of Essex, Colchester, United Kingdom griffm@essex.ac.uk Received: /19/16, Accepted:
More informationOn the Shifted Product of Binary Recurrences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), rticle 10.6.1 On the Shifted Product of Binary Recurrences Omar Khadir epartment of Mathematics University of Hassan II Mohammedia, Morocco
More informationOn the indecomposability of polynomials
On the indecomposability of polynomials Andrej Dujella, Ivica Gusić and Robert F. Tichy Abstract Applying a combinatorial lemma a new sufficient condition for the indecomposability of integer polynomials
More informationDeterminant and Permanent of Hessenberg Matrix and Fibonacci Type Numbers
Gen. Math. Notes, Vol. 9, No. 2, April 2012, pp.32-41 ISSN 2219-7184; Copyright c ICSRS Publication, 2012 www.i-csrs.org Available free online at http://www.geman.in Determinant and Permanent of Hessenberg
More informationHomework #2 solutions Due: June 15, 2012
All of the following exercises are based on the material in the handout on integers found on the class website. 1. Find d = gcd(475, 385) and express it as a linear combination of 475 and 385. That is
More informationGeneralized Lucas Sequences Part II
Introduction Generalized Lucas Sequences Part II Daryl DeFord Washington State University February 4, 2013 Introduction Èdouard Lucas: The theory of recurrent sequences is an inexhaustible mine which contains
More informationOn The Discriminator of Lucas Sequences
On The Discriminator of Lucas Sequences Bernadette Faye Ph.D Student FraZA, Bordeaux November 8, 2017 The Discriminator Let a = {a n } n 1 be a sequence of distinct integers. The Discriminator is defined
More informationk-jacobsthal and k-jacobsthal Lucas Matrix Sequences
International Mathematical Forum, Vol 11, 016, no 3, 145-154 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/imf0165119 k-jacobsthal and k-jacobsthal Lucas Matrix Sequences S Uygun 1 and H Eldogan Department
More informationAND OCCUPANCY PROBLEMS
COMBINATIONS, COMPOSITIONS AND OCCUPANCY PROBLEMS MORTON ABRAMSON York University, Downsview, Toronto, Canada Let r < k be positive integers,, INTRODUCTION By a composition of k into r parts (an r-composition
More informationImpulse Response Sequences and Construction of Number Sequence Identities
Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL 6170-900, USA Abstract As an extension of Lucas
More informationNewton, Fermat, and Exactly Realizable Sequences
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.2 Newton, Fermat, and Exactly Realizable Sequences Bau-Sen Du Institute of Mathematics Academia Sinica Taipei 115 TAIWAN mabsdu@sinica.edu.tw
More informationInvestigating Geometric and Exponential Polynomials with Euler-Seidel Matrices
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.6 Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices Ayhan Dil and Veli Kurt Department of Mathematics
More informationSums of Squares and Products of Jacobsthal Numbers
1 2 47 6 2 11 Journal of Integer Sequences, Vol. 10 2007, Article 07.2.5 Sums of Squares and Products of Jacobsthal Numbers Zvonko Čerin Department of Mathematics University of Zagreb Bijenička 0 Zagreb
More information5. Sequences & Recursion
5. Sequences & Recursion Terence Sim 1 / 42 A mathematician, like a painter or poet, is a maker of patterns. Reading Sections 5.1 5.4, 5.6 5.8 of Epp. Section 2.10 of Campbell. Godfrey Harold Hardy, 1877
More information