Gaussian Pell Numbers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
ON SOME GAUSSIAN PELL and PELL-LUCAS NUMBERS Abstract
Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:1, 2016,8-18/Ordu Univ. J. Sci. Tech., Vol:6, No:1,2016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı*1 Sinan Öz2 1Pamukkale Uni., Science and Arts Faculty,Dept. of Math., KınıklıCampus, Denizli, Turkey 2Pamukkale Uni., Science and Arts Faculty,Dept. of Math., Denizli, Turkey Abstract In this study, we consider firstly the generalized Gaussian Fibonacci and Lucas sequences. Then we define the Gaussian Pell and Gaussian Pell-Lucas sequences. We give the generating functions and Binet formulas of Gaussian Pell and Gaussian Pell- Lucas sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Keywords. Recurrence Relation, Fibonacci numbers, Gaussian Pell and Pell-Lucas numbers. Özet Bu çalışmada, önce genelleştirilmiş Gaussian Fibonacci ve Lucas dizilerini dikkate aldık. Sonra, Gaussian Pell ve Gaussian Pell-Lucas dizilerini tanımladık. Gaussian Pell ve Gaussian Pell-Lucas dizilerinin Binet formüllerini ve üreteç fonksiyonlarını verdik. Üstelik, Gaussian Pell ve Gaussian Pell-Lucas sayılarını içeren bazı önemli özdeşlikler elde ettik. AMS Classification. 11B37, 11B39.1 * [email protected], 8 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well known Generalized Fibonacci sequence {푈푛}, 푈푛+1 = 푝푈푛 + 푞푈푛−1 , 푈0 = 0 푎푛푑 푈1 = 1, and generalized Lucas sequence {푉푛} are defined by 푉푛+1 = 푝푉푛 + 푞푉푛−1 , 푉0 = 2 푎푛푑 푉1 = 푝, where 푝 and 푞 are nonzero real numbers and 푛 ≥ 1. For 푝 = 푞 = 1, we have classical Fibonacci and Lucas sequences. For 푝 = 2, 푞 = 1, we have Pell and Pell- Lucas sequences. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Twelve Simple Algorithms to Compute Fibonacci Numbers Arxiv
Twelve Simple Algorithms to Compute Fibonacci Numbers Ali Dasdan KD Consulting Saratoga, CA, USA [email protected] April 16, 2018 Abstract The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used in introductory algorithms courses. In this paper, we present twelve of these well-known algo- rithms and some of their properties. These algorithms, though very simple, illustrate multiple concepts from the algorithms field, so we highlight them. We also present the results of a small-scale experi- mental comparison of their runtimes on a personal laptop. Finally, we provide a list of homework questions for the students. We hope that this paper can serve as a useful resource for the students learning the basics of algorithms. arXiv:1803.07199v2 [cs.DS] 13 Apr 2018 1 Introduction The Fibonacci numbers are a sequence Fn of integers in which every num- ber after the first two, 0 and 1, is the sum of the two preceding num- bers: 0; 1; 1; 2; 3; 5; 8; 13; 21; ::. More formally, they are defined by the re- currence relation Fn = Fn−1 + Fn−2, n ≥ 2 with the base values F0 = 0 and F1 = 1 [1, 5, 7, 8]. 1 The formal definition of this sequence directly maps to an algorithm to compute the nth Fibonacci number Fn. However, there are many other ways of computing the nth Fibonacci number. -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Fibonacci Numbers
mathematics Article On (k, p)-Fibonacci Numbers Natalia Bednarz The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, al. Powsta´nców Warszawy 12, 35-959 Rzeszów, Poland; [email protected] Abstract: In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultane- ously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too. Keywords: Fibonacci numbers; Pell numbers; Narayana numbers MSC: 11B39; 11B83; 11C20 1. Introduction By numbers of the Fibonacci type we mean numbers defined recursively by the r-th order linear recurrence relation of the form an = b1an−1 + b2an−2 + ··· + bran−r, for n > r, (1) where r > 2 and bi > 0, i = 1, 2, ··· , r are integers. For special values of r and bi, i = 1, 2, ··· r, the Equality (1) defines well-known numbers of the Fibonacci type and their generalizations. We list some of them: Citation: Bednarz, N. On 1. Fibonacci numbers: Fn = Fn−1 + Fn−2 for n > 2, with F0 = F1 = 1. (k, p)-Fibonacci Numbers. 2. Lucas numbers: Ln = Ln−1 + Ln−2 for n > 2, with L0 = 2, L1 = 1. Mathematics 2021, 9, 727. https:// 3. Pell numbers: Pn = 2Pn−1 + Pn−2 for n > 2, with P0 = 0, P1 = 1. doi.org/10.3390/math9070727 4. Pell–Lucas numbers: Qn = 2Qn−1 + Qn−2 for n > 2, with Q0 = 1, Q1 = 3. 5. Jacobsthal numbers: Jn = Jn−1 + 2Jn−2 for n > 2, with J0 = 0, J1 = 1. -
On Repdigits As Product of Consecutive Fibonacci Numbers1
Rend. Istit. Mat. Univ. Trieste Volume 44 (2012), 393–397 On repdigits as product of consecutive Fibonacci numbers1 Diego Marques and Alain Togbe´ Abstract. Let (Fn)n≥0 be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn ··· Fn+(k−1) is a repdigit, with at least two digits, then (k, n) = (1, 10). Keywords: Fibonacci, repdigits, sequences (mod m) MS Classification 2010: 11A63, 11B39, 11B50 1. Introduction Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2 = Fn+1 + Fn, for n ≥ 0, where F0 = 0 and F1 = 1. These numbers are well-known for possessing amaz- ing properties. In 1963, the Fibonacci Association was created to provide an opportunity to share ideas about these intriguing numbers and their applica- tions. We remark that, in 2003, Bugeaud et al. [2] proved that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 (see [6] for the Fibono- mial version). In 2005, Luca and Shorey [5] showed, among other things, that a non-zero product of two or more consecutive Fibonacci numbers is never a perfect power except for the trivial case F1 · F2 = 1. Recall that a positive integer is called a repdigit if it has only one distinct digit in its decimal expansion. In particular, such a number has the form a(10m − 1)/9, for some m ≥ 1 and 1 ≤ a ≤ 9. -
Adopting Number Sequences for Shielding Information
Manish Bansal et al, International Journal of Computer Science and Mobile Computing, Vol.3 Issue.11, November- 2014, pg. 660-667 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 3, Issue. 11, November 2014, pg.660 – 667 RESEARCH ARTICLE Adopting Number Sequences for Shielding Information Mrs. Sandhya Maitra, Mr. Manish Bansal, Ms. Preety Gupta Associate Professor - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India [email protected] , [email protected] , [email protected] Abstract:-The advancement of technology and global communication networks puts up the question of safety of conveyed data and saved data over these media. Cryptography is the most efficient and feasible mode to transfer security services and also Cryptography is becoming effective tool in numerous applications for information security. This paper studies the shielding of information with the help of cryptographic function and number sequences. The efficiency of the given method is examined, which assures upgraded cryptographic services in physical signal and it is also agile and clear for realization. 1. Introduction Cryptography is the branch of information security. The basic goal of cryptography is to make communication secure in the presence of third party (intruder). Encryption is the mechanism of transforming information into unreadable (cipher) form. This is accomplished for data integrity, forward secrecy, authentication and mainly for data confidentiality. -
On the Connections Between Pell Numbers and Fibonacci P-Numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 27, 2021, No. 1, 148–160 DOI: 10.7546/nntdm.2021.27.1.148-160 On the connections between Pell numbers and Fibonacci p-numbers Anthony G. Shannon1, Ozg¨ ur¨ Erdag˘2 and Om¨ ur¨ Deveci3 1 Warrane College, University of New South Wales Kensington, Australia e-mail: [email protected] 2 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] 3 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] Received: 24 April 2020 Revised: 4 January 2021 Accepted: 7 January 2021 Abstract: In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices. Keywords: Pell sequence, Fibonacci p-sequence, Matrix, Representation. 2010 Mathematics Subject Classification: 11K31, 11C20, 15A15. 1 Introduction The well-known Pell sequence fPng is defined by the following recurrence relation: Pn+2 = 2Pn+1 + Pn for n ≥ 0 in which P0 = 0 and P1 = 1. 148 There are many important generalizations of the Fibonacci sequence. The Fibonacci p-sequence [22, 23] is one of them: Fp (n) = Fp (n − 1) + Fp (n − p − 1) for p = 1; 2; 3;::: and n > p in which Fp (0) = 0, Fp (1) = ··· = Fp (p) = 1. -
Product of Consecutive Tribonacci Numbers with Only One Distinct Digit
1 2 Journal of Integer Sequences, Vol. 22 (2019), 3 Article 19.6.3 47 6 23 11 Product of Consecutive Tribonacci Numbers With Only One Distinct Digit Eric F. Bravo and Carlos A. G´omez Department of Mathematics Universidad del Valle Calle 13 No 100 – 00 Cali Colombia [email protected] [email protected] Florian Luca School of Mathematics University of the Witwatersrand Johannesburg South Africa and Research Group in Algebraic Structures and Applications King Abdulaziz University Jeddah Saudi Arabia and Department of Mathematics University of Ostrava 30 Dubna 22, 701 03 Ostrava 1 Czech Republic [email protected] Abstract Let (Fn)n≥0 be the sequence of Fibonacci numbers. Marques and Togb´eproved that if the product Fn ··· Fn+ℓ−1 is a repdigit (i.e., a number with only distinct digit 1 in its decimal expansion), with at least two digits, then (ℓ,n)=(1, 10). In this paper, we solve the same problem with Tribonacci numbers instead of Fibonacci numbers. 1 Introduction A positive integer is called a repdigit if it has only one distinct digit in its decimal expansion. The sequence of numbers with repeated digits is included in Sloane’s On-Line Encyclopedia of Integer Sequences (OEIS) [11] as sequence A010785. In 2000, Luca [6] showed that the largest repdigit Fibonacci number is F10 = 55 and the largest repdigit Lucas number is L5 = 11. Motivated by the results of Luca [6], several authors have explored repdigits in general- izations of Fibonacci numbers and Lucas numbers. For instance, Bravo and Luca [1] showed 1 (3) that the only repdigit in the k-generalized Fibonacci sequence , is F8 = 44. -
Retrograde Renegades and the Pascal Connection: Repeating Decimals Represented by Fibonacci and Other Sequences Appearing from Right to Left
RETROGRADE RENEGADES AND THE PASCAL CONNECTION: REPEATING DECIMALS REPRESENTED BY FIBONACCI AND OTHER SEQUENCES APPEARING FROM RIGHT TO LEFT Marjorie Bicknell-Johnson Santa Clara Unified School District, Santa Clara, CA 95051 (Submitted October 1987) Repeating decimals show a surprisingly rich variety of number sequence patterns when their repetends are viewed in retrograde fashion, reading from the rightmost digit of the repeating cycle towards the left. They contain geometric sequences as well as Fibonacci numbers generated by an application of Pascal*s triangle. Further, fractions whose repetends end with successive terms of Fnm , m = 1, 2, ..., occurring in repeating blocks of k digits, are completely characterized, as well as fractions ending with Fnm+p or Lnm+p, th where Fn is the n Fibonacci number, F1 = 1, F2 = 1, Fn+i = Fn + Fn_19 th and Ln is the n Lucas number, 1. The Pascal Connection It is no surprise that 1/89 contains the sum of successive Fibonacci num- bers in its decimal expansion [2], [3], [4], [5], as 1/89 = .012358 13 21 34 However, 1/89 can also be expressed as the sum of successive powers of 11, as 1/89 = .01 .0011 .000121 .00001331 .0000014641 where 1/89 = 1/102 + ll/lO4 + 1'12/106 + ..., which is easily shown by summing the geometric progression. If the array above had the leading zeros removed and was left-justified, we would have Pascal's triangle in a form where the Fibonacci numbers arise as the sum of the rising diagonals. Notice that llk generates rows of Pascal's triangle, and that the columns of the array expressing 1/89 are the diagonals of Pascal's triangle. -
A NOTE on BICOMPLEX FIBONACCI and LUCAS NUMBERS Semra
International Journal of Pure and Applied Mathematics Volume 120 No. 3 2018, 365-377 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v120i3.7 ijpam.eu A NOTE ON BICOMPLEX FIBONACCI AND LUCAS NUMBERS Semra Kaya Nurkan1 §, Ilkay˙ Arslan G¨uven2 1Department of Mathematics U¸sak University 64200, U¸sak, TURKEY 2Department of Mathematics Gaziantep University 27310, Gaziantep, TURKEY Abstract: In this study, we define a new type of Fibonacci and Lucas numbers which are called bicomplex Fibonacci and bicomplex Lucas numbers. We obtain the well-known properties e.g. D’ocagnes, Cassini, Catalan for these new types. We also give the identities of negabicomplex Fibonacci and negabicomplex Lucas numbers, Binet formulas and relations of them. AMS Subject Classification: 11B39, 11E88, 11R52 Key Words: Fibonacci numbers, Lucas numbers, bicomplex numbers 1. Introduction Fibonacci numbers are invented by Italian mathematician Leonardo Fibonacci when he wrote his first book Liber Abaci in 1202 contains many elementary problems, including famous rabbit problem. The Fibonacci numbers are the numbers of the following integer sequence; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... The sequence of Fibonacci numbers which is denoted by Fn is defined as the Received: February 21, 2017 c 2018 Academic Publications, Ltd. Revised: March 28, 2018 url: www.acadpubl.eu Published: December 4, 2018 §Correspondence author 366 S.K. Nurkan, I.A.˙ G¨uven linear reccurence relation Fn = Fn−1 + Fn−2 with F = F = 1 and n Z. Also as a result of this relation we can define F = 1 2 ∈ 0 0. -
Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and Their Explicit Formulas by Ian Walker Ju
Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and their Explicit Formulas By Ian Walker June 2011 MST 501 Research Project In partial fulfillment for the Master’s of Teaching Mathematics (M.S.T.) Portland State University, Portland, Oregon John Pell (1611-1685) 2 John Pell (1611-1685) An “obscure” English Mathematician • Part of the 17th century intellectual history of England and of Continental Europe. • Pell was married with eight children, taught math at the Gymnasium in Amsterdam, and was Oliver Cromwell’s envoy to Switzerland. • Pell was well read in classical and contemporary mathematics. • Pell had correspondence with Descartes, Leibniz, Cavendish, Mersenne, Hartlib, Collins and others. • His main mathematical focus was on mathematical tables: tables of squares, sums of squares, primes and composites, constant differences, logarithms, antilogarithms, trigonometric functions, etc. 3 John Pell (1611-1685) An “obscure” English Mathematician • Many of Pell’s booklets of tables and other works do not list himself as the author. • Did not publish much mathematical work. Is more known for his activities, correspondence and contacts. • Only one of his tables was ever published (1672), which had tables of the first 10,000 square numbers. • His best known published work is, “An Introduction to Algebra”. It explains how to simplify and solve equations. • Pell is credited with the modern day division symbol and the double-angle tangent formula. • Pell is best known, only by name, for the Pell Sequence and the Pell Equation. 4 John Pell (1611-1685) An “obscure” English Mathematician • Division Symbol: • Double-Angle Tangent Formula: 2tan tan2 2 • Pell Sequence: 1 tan p 2p p p 1, p 2,n 2 n n1 n2 0 1 • Pell Equation: 2 2 x 2y 1 5 John Pell (1611-1685) An “obscure” English Mathematician • Both the Pell Sequence and the Pell Equation are erroneously named after him.