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On the period mod m of polynomially-recursive sequences: a case study [article]

Cyril Banderier, Florian Luca
2019 arXiv   pre-print
Polynomially-recursive sequences generally have a periodic behavior mod m. In this paper, we analyze the period mod m of a second order polynomially-recursive sequence.  ...  We give the mod a^k supercongruences, and generalize these results to a class of recurrences.  ...  In addition, Florian Luca was supported in part by grant CPRR160325161141 and an A-rated scientist award both from the NRF of South Africa and by grant no. 17-02804S of the Czech Granting Agency.  ... 
arXiv:1903.01986v2 fatcat:dehe6q3cgjap3j4a5dpwg7b3yq

On polynomial recursive sequences [article]

Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, Géraud Sénizergues
2020 arXiv   pre-print
We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences.  ...  A typical example of a polynomial recursive sequence is b_n=n!. Our main result is that the sequence u_n=n^n is not polynomial recursive.  ...  Finally, we thank the participants of the automata seminar at the University of Warsaw for an insightful discussion on the class of rational recursive sequences (considered in Section 7).  ... 
arXiv:2002.08630v1 fatcat:mle5zy7lgbhotmcrnl63ywteyu

On Polynomial Recursive Sequences

Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, Géraud Sénizergues, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences.  ...  A typical example of a polynomial recursive sequence is b_n = n!. Our main result is that the sequence u_n = nⁿ is not polynomial recursive.  ...  The goal of this paper is to study the expressive power of polynomial recursive sequences.  ... 
doi:10.4230/lipics.icalp.2020.117 dblp:conf/icalp/CadilhacMPPS20 fatcat:vye7gwvtqvdrpabndnnko23q5a

On Polynomial Recursive Sequences

Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, Géraud Sénizergues
2021 Theory of Computing Systems  
AbstractWe study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences.  ...  A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.  ...  Finally, we thank the participants of the automata seminar at the University of Warsaw for an insightful discussion on the class of rational recursive sequences.  ... 
doi:10.1007/s00224-021-10046-9 fatcat:6ruy4kktmvbtnkeb666kf2r6q4

Pseudonoise Sequences [chapter]

Tor Helleseth, P Kumar
2012 Mobile Communications Handbook, Third Edition  
For example, this is true in the case of m sequences (the remaining moments turn out to depend upon the specific characteristic polynomial of the m sequence).  ...  A sequence satisfying a recursion of the form in Eq. (8.1) is said to have characteristic polynomial f (x) = r i=0 f i x i .  ... 
doi:10.1201/b12494-14 fatcat:vjtj2xlm4nf6pejp74dtyxii2i

Pseudonoise Sequences [chapter]

Tor Helleseth, P Kumar
2002 The Communications Handbook  
For example, this is true in the case of m sequences (the remaining moments turn out to depend upon the specific characteristic polynomial of the m sequence).  ...  A sequence satisfying a recursion of the form in Eq. (8.1) is said to have characteristic polynomial f (x) = r i=0 f i x i .  ... 
doi:10.1201/9781420041163-10 fatcat:cd7djnz4fjbgjl2a3kxx2n6r3u

Pseudonoise Sequences [chapter]

Tor Helleseth, P Vijay Kumar
1999 Electrical Engineering Handbook  
For example, this is true in the case of m sequences (the remaining moments turn out to depend upon the specific characteristic polynomial of the m sequence).  ...  A sequence satisfying a recursion of the form in Eq. (8.1) is said to have characteristic polynomial f (x) = r i=0 f i x i .  ... 
doi:10.1201/noe0849321672.ch8 fatcat:ksn6t4z7izfn3c6ppu5t4q67ji

Recursive sequences and polynomial congruences

J. Lehman, Christopher Triola
2010 Involve. A Journal of Mathematics  
This research started on a National Science Foundation Research Experience for Undergraduates Program at the Rochester Institute of Technology (RIT) during the summer of 2007, which was cofunded by Department  ...  Acknowledgments The authors thank the anonymous referee for advice on how to improve the clarity of the paper.  ...  Periodicity of recursive sequences modulo integers Let m be a positive integer.  ... 
doi:10.2140/involve.2010.3.129 fatcat:57bb726t2bfojnudqlczwkatie

On Sequence Groups [article]

Zbigniew Lipinski, Maciej P. Wojtkowski
2021 arXiv   pre-print
Linear second order recursive sequences with arbitrary initial conditions are studied.  ...  Prime divisors of sequences are studied with the help of the sequence group p, which is always cyclic of order p± 1.  ...  In Section 6 we proceed with the study of the sets of prime divisors of recursive sequences, mostly in the language of one rational parameter t.  ... 
arXiv:2110.00450v1 fatcat:nf5lj6ret5ap5og36mkw6clgly

On aperiodic and periodic complementary binary sequences

K. Feng, P. Jau-Shyong, Qing Xiang
1999 IEEE Transactions on Information Theory  
We give direct and recursive constructions for aperiodic and periodic complementary sequences. Using these constructions, many missing entries in the table of Bömer and Antweiler [4] can be filled.  ...  The research of Qing Xiang was partially supported by NSA grant MDA 904-97-0104.  ...  Peter Jau-Shyong Shiue would like to thank UNLV for a sabbatical leave during which the research in this paper was carried out.  ... 
doi:10.1109/18.746823 fatcat:oa42hibtvbhbxbmd6kjou2kt6u

Lacunary formal power series and the Stern-Brocot sequence [article]

Jean-Paul Allouche, Michel Mendès France
2013 arXiv   pre-print
of the Stern-Brocot sequence.  ...  In all the other cases Q_ω(X) is an infinite formal power series, the algebraic properties of which we discuss in the special case λ_n = 2^n+1 - 1.  ...  Quite obviously, in the case of the above F (X), one has A 0 (X) = 0. Let P n (X)/Q n (X) = [0, A 1 (X), A 2 (X), ..., A n (X)] be the nth convergent of F (X).  ... 
arXiv:1202.0211v3 fatcat:svkaui2barhnxneofntdydbkke

Crosscorrelation of M-sequences and GMW-sequences with the same primitive polynomial

Richard A. Games
1985 Discrete Applied Mathematics  
The periodic crosscorrelation function of an M-sequence and a GMW-sequence that have been constructed from the same primitive polynomial is calculated in terms of the perodic crosscorrelation function  ...  of two smaller M-sequences.  ...  In the case of GF (2) , the sequences obtained from Singer difference sets correspond to binary maximum period linear recursive sequences, or M-sequences [3] , [4] .  ... 
doi:10.1016/0166-218x(85)90067-8 fatcat:bdbtjklxrvb2dkdcst3ckx3z44

p-adic asymptotic properties of constant-recursive sequences

Eric Rowland, Reem Yassawi
2017 Indagationes mathematicae  
The second is that the density of the residues modulo p^α attained by a constant-recursive sequence converges, as α→∞, to the Haar measure of a certain subset of Z_p.  ...  In this article we study p-adic properties of sequences of integers (or p-adic integers) that satisfy a linear recurrence with constant coefficients.  ...  Acknowledgments The authors thank Valérie Berthé for helpful discussions. The second author thanks LIAFA, Université Paris-7 for its hospitality and support.  ... 
doi:10.1016/j.indag.2016.11.019 fatcat:prp6kc6pxngppiwmdwuht6wsfe

A generalized Lucas sequence and permutation binomials

Amir Akbary, Qiang Wang
2005 Proceedings of the American Mathematical Society  
By employing the integer sequence a n = l−1 2 t=1 2 cos π(2t − 1) l n , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials P (x) = x r + x u of the finite  ...  Let p be an odd prime and q = p m . Let l be an odd positive integer. Let p ≡ −1 (mod l) or p ≡ 1 (mod l) and l | m.  ...  Acknowledgment The authors would like to thank the referee for helpful comments and suggestions.  ... 
doi:10.1090/s0002-9939-05-08220-1 fatcat:qz77vzt7f5b6fihuizbxm5zznq

Transform Domain Analysis of Sequences [article]

Muhammad Asad Khan, Amir A Khan, Fauzan Mirza
2015 arXiv   pre-print
Motivated by DFT based attacks, we present a transform domain analysis of Linear Feedback Shift Register(LFSR) based sequence generators.  ...  CRT is exploited to establish patterns in LFSR sequences and underlying cyclic structures of finite fields. Application of DFT spectra attacks on combiner generators is also demonstrated.  ...  The period of output stream z t is 2 m − 1 where m is degree of feedback polynomial of LFSR which is primitive in most cases.  ... 
arXiv:1503.00943v2 fatcat:cjrh22b6tfbdtdr5id5rmguuau
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