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

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. ...

##
###
On polynomial recursive sequences
[article]

2020
*
arXiv
*
pre-print

We

arXiv:2002.08630v1
fatcat:mle5zy7lgbhotmcrnl63ywteyu
*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). ...##
###
On Polynomial Recursive Sequences

2020
*
International Colloquium on Automata, Languages and Programming
*

We

doi:10.4230/lipics.icalp.2020.117
dblp:conf/icalp/CadilhacMPPS20
fatcat:vye7gwvtqvdrpabndnnko23q5a
*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*. ...##
###
On Polynomial Recursive Sequences

2021
*
Theory of Computing Systems
*

AbstractWe

doi:10.1007/s00224-021-10046-9
fatcat:6ruy4kktmvbtnkeb666kf2r6q4
*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*. ...##
###
Pseudonoise Sequences
[chapter]

2012
*
Mobile Communications Handbook, Third Edition
*

For example, this is true in

doi:10.1201/b12494-14
fatcat:vjtj2xlm4nf6pejp74dtyxii2i
*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 . ...##
###
Pseudonoise Sequences
[chapter]

2002
*
The Communications Handbook
*

For example, this is true in

doi:10.1201/9781420041163-10
fatcat:cd7djnz4fjbgjl2a3kxx2n6r3u
*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 . ...##
###
Pseudonoise Sequences
[chapter]

1999
*
Electrical Engineering Handbook
*

For example, this is true in

doi:10.1201/noe0849321672.ch8
fatcat:ksn6t4z7izfn3c6ppu5t4q67ji
*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 . ...##
###
Recursive sequences and polynomial congruences

2010
*
Involve. A Journal of Mathematics
*

This research started

doi:10.2140/involve.2010.3.129
fatcat:57bb726t2bfojnudqlczwkatie
*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. ...##
###
On Sequence Groups
[article]

2021
*
arXiv
*
pre-print

Linear second order

arXiv:2110.00450v1
fatcat:nf5lj6ret5ap5og36mkw6clgly
*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. ...##
###
On aperiodic and periodic complementary binary sequences

1999
*
IEEE Transactions on Information Theory
*

We give direct and

doi:10.1109/18.746823
fatcat:oa42hibtvbhbxbmd6kjou2kt6u
*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. ...##
###
Lacunary formal power series and the Stern-Brocot sequence
[article]

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). ...

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

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] . ...

##
###
p-adic asymptotic properties of constant-recursive sequences

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. ...

##
###
A generalized Lucas sequence and permutation binomials

2005
*
Proceedings of the American Mathematical Society
*

By employing

doi:10.1090/s0002-9939-05-08220-1
fatcat:qz77vzt7f5b6fihuizbxm5zznq
*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. ...##
###
Transform Domain Analysis of Sequences
[article]

2015
*
arXiv
*
pre-print

Motivated by DFT based attacks, we present

arXiv:1503.00943v2
fatcat:cjrh22b6tfbdtdr5id5rmguuau
*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*. ...
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