The ring of k-regular sequences, II

In this paper, we continue our study of k-regular sequences begun in 1992. We prove some new results, give many new examples from the literature, and state some open problems.

[1]  Jean Berstel,et al.  Rational series and their languages , 1988, EATCS monographs on theoretical computer science.

[2]  G. Rauzy.  Suites à termes dans un alphabet fini , 1983 .

[3]  J. Lambek,et al.  On Some two way Classifications of Integers , 1959, Canadian Mathematical Bulletin.

[4]  Marcel Paul Schützenberger,et al.  On the Definition of a Family of Automata , 1961, Inf. Control..

[5]  Alfred J. van der Poorten,et al.  From Geometry to Euler Identities , 1989, Theor. Comput. Sci..

[6]  A. Pfister,et al.  Quadratische Formen in beliebigen Körpern , 1966 .

[7]  Théodore Tapsoba.  Complexité de suites automatiques , 1987 .

[8]  P. Shiu,et al.  A Correlated Digital Sum Problem Associated with Sums of Three Squares , 1989 .

[9]  K. Mahler,et al.  On a Special Functional Equation , 1940 .

[10]  R. Bellman,et al.  On a Problem in Additive Number Theory , 1948 .

[11]  de Ng Dick Bruijn.  On Mahler's partition problem , 1948 .

[12]  Martin D. Davis.  Hilbert's Tenth Problem is Unsolvable , 1973 .

[13]  Aline Weitzman.  Transformation of Parallel Programs Guided by Micro-Analysis , 1992 .

[14]  B. Mossé.  Reconnaissabilité des substitutions et complexité des suites automatiques , 1996 .

[15]  Philippe Flajolet,et al.  A Note on Gray Code and Odd-Even Merge , 1980, SIAM J. Comput..

[16]  Donald E. Knuth,et al.  Sorting and Searching , 1973 .

[17]  Vijay K. Varadan,et al.  Fractal sequences derived from the self-similar extensions of the Sierpinski gasket , 1988 .

[18]  Donald J. Newman,et al.  On the number of binary digits in a multiple of three , 1969 .

[19]  Patrick Morton,et al.  Paper Folding, Digit Patterns and Groups of Arithmetic Fractals , 1989 .

[20]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[21]  Arto Salomaa,et al.  Automata-Theoretic Aspects of Formal Power Series , 1978, Texts and Monographs in Computer Science.

[22]  M. Keane,et al.  0-1-sequences of Toeplitz type , 1969 .

[23]  Tsit Yuen Lam,et al.  The algebraic theory of quadratic forms , 1973 .

[24]  Jean-Paul Allouche.  Somme des chiffres et transcendance , 1982 .

[25]  de Ng Dick Bruijn.  Some direct decompositions of the set on integers , 1964 .

[26]  E. A. Maxwell,et al.  Mathematical Gems II , 1976, The Mathematical Gazette.

[27]  Clark Kimberling,et al.  Numeration systems and fractal sequences , 1995 .

[28]  Jeffrey Shallit,et al.  On integer sequences whose first iterates are linear , 2005 .

[29]  Jeffrey Shallit,et al.  The Ring of k-Regular Sequences , 1990, Theor. Comput. Sci..

[30]  M. Stern.  Ueber eine zahlentheoretische Funktion. , 1858 .

[31]  Jeffrey Shallit,et al.  Pattern Spectra, Substring Enumeration, and Automatic Sequences , 1992, Theor. Comput. Sci..

[32]  Leo Moser.  An Application of Generating Series , 1962 .

[33]  Marcel Paul Schützenberger,et al.  Counting with Rational Functions , 1986, Theor. Comput. Sci..

[34]  Albrecht Pfjster,et al.  Zur Darstellung von -1 Als Summe von Quadraten in Einem Körper , 1965 .

[35]  L. Carlitz.  Single variable bell polynomials , 1962 .

[36]  J. M. Dubbey.  The Mathematical Work of Charles Babbage , 1978 .

[37]  S. Ginsburg,et al.  BOUNDED ALGOL-LIKE LANGUAGES^) , 1964 .

[38]  Neil J. A. Sloane,et al.  The encyclopedia of integer sequences , 1995 .

[39]  A. Salzberg,et al.  Exact Statistical Thermodynamics of Gravitational Interactions in One and Two Dimensions , 1965 .

[40]  Jean-Paul Allouche,et al.  Palindrome complexity , 2003, Theor. Comput. Sci..

[41]  Jean-Paul Allouche,et al.  Note sur un article de Sharif et Woodcock , 1989 .

[42]  László Babai.  On the length of subgroup chains in the symmetric group , 1986 .

[43]  Michael A. Harrison,et al.  Lectures on linear sequential machines , 1969 .

[44]  Helmut Prodinger.  Non-repetitive sequences and gray code , 1983, Discret. Math..

[45]  K. Mahler,et al.  The Spectrum of an Array and its Application to the Study of the Translation Properties of a Simple Class of Arithmetical Functions: Part Two On the Translation Properties of a Simple Class of Arithmetical Functions , 1927 .

[46]  Jeffrey Shallit,et al.  The Ubiquitous Prouhet-Thue-Morse Sequence , 1998, SETA.

[47]  Alfred J. van der Poorten,et al.  An awful problem about integers in base four , 1987 .

[48]  J. Shallit,et al.  Wire bending , 1989, J. Comb. Theory, Ser. A.

[49]  Jeffrey Shallit,et al.  THE MATHEMATICS OF PER NØRGÅRD ’ S RHYTHMIC INFINITY SYSTEM , 2002 .

[50]  Jeffrey Shallit,et al.  Advanced Problems: 6625-6627 , 1990 .

[51]  Aldo de Luca,et al.  Some Combinatorial Properties of the Thue-Morse Sequence and a Problem in Semigroups , 1989, Theor. Comput. Sci..

[52]  Edsger W. Dijkstra,et al.  Selected Writings on Computing: A personal Perspective , 1982, Texts and Monographs in Computer Science.

[53]  Olivier Salon,et al.  Quelles tuiles ! (Pavages apériodiques du plan et automates bidimensionnels) , 1989 .

[54]  S. Ginsburg,et al.  Bounded -like languages , 1964 .

[55]  Julien Cassaigne,et al.  Counting Overlap-Free Binary Words , 1993, STACS.

[56]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[57]  Akhlesh Lakhtakia,et al.  Self-similar sequences and chaos from Gauss sums , 1989, Comput. Graph..

[58]  Donald J. Newman,et al.  Binary digit distribution over naturally defined sequences , 1975 .

[59]  Peter J. Cameron,et al.  Chains of subgroups in symmetric groups , 1989 .

[60]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[61]  Joe Roberts.  Lure of the Integers , 1992 .

[62]  Alan Cobham,et al.  On the base-dependence of sets of numbers recognizable by finite automata , 1969, Mathematical systems theory.

[63]  Srecko Brlek,et al.  Enumeration of factors in the Thue-Morse word , 1989, Discret. Appl. Math..

[64]  P. Shiu.  Counting Sums of Three Squares , 1988 .

[65]  R. Guy.  Unsolved Problems in Number Theory , 1981 .

[66]  Arthur Porges.  A Set of Eight Numbers , 1945 .

[67]  Vijay K. Varadan,et al.  COMMENT: On a new class of planar fractals: the Pascal-Sierpinski gaskets , 1986 .

[68]  J. Coquet,et al.  A summation formula related to the binary digits , 1983 .

[69]  Michel Dekking,et al.  Folds: III. More Morphisms , 1982 .

[70]  B. Reznick.  Some Binary Partition Functions , 1990 .

[71]  B. M. Stewart.  Sums of Functions of Digits , 1960, Canadian Journal of Mathematics.

[72]  J. Conway.  On Numbers and Games , 1976 .

[73]  P Cheo,et al.  A PROBLEM ON THE k-ADIC REPRESENTATION OF POSITIVE INTEGERS , 1955 .

[74]  Alan Cobham,et al.  Uniform tag sequences , 1972, Mathematical systems theory.

[75]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[76]  Robert E. Tarjan,et al.  On a Greedy Heuristic for Complete Matching , 1981, SIAM J. Comput..

[77]  Shi-Chun Tsai,et al.  Exact solution of a minimal recurrence , 2000, Inf. Process. Lett..

[78]  L. Carlitz.  A problem in partitions related to the Stirling numbers , 1964 .

[79]  A. J. Poorten,et al.  Arithmetic properties of automata: regular sequences. , 1988 .

[80]  G. Rauzy,et al.  Suites algébriques, automates et substitutions , 1980 .

[81]  Jeffrey Shallit,et al.  Regular Expressions: New Results and Open Problems , 2004, J. Autom. Lang. Comb..

[82]  Samuel S. Wagstaff.  The Schnirelmann density of the sums of three squares , 1975 .

[83]  Olivier Salon.  Propriétés arithmétiques des automates multidimensionnels , 1989 .

[84]  Jeffrey Shallit,et al.  Automatic Sequences by Jean-Paul Allouche , 2003 .

[85]  N. J. Fine,et al.  Binomial Coefficients Modulo a Prime , 1947 .

[86]  Paul Erdös,et al.  On Some Sequences of Integers , 1936 .

[87]  Bruce Reznick.  Some extremal problems for continued fractions , 1985 .

[88]  L. Carlitz,et al.  Some partition problems related to the Stirling numbers of the second kind , 1965 .

[89]  D. H. Lehmer,et al.  Integers with Digits 0 or 1 , 1986 .

[90]  Richard J. Lipton,et al.  Word Problems Solvable in Logspace , 1977, JACM.

[91]  Andrew Chi-Chih Yao,et al.  Addition chains with multiplicative cost , 1978, Discret. Math..

[92]  G. F. Clements,et al.  A sequence of (±1)-determinants with large values , 1965 .

[93]  Charles Babbage,et al.  Babbage's Calculating Engines: Observations on the Application of Machinery to the Computation of Mathematical Tables , 1825 .

[94]  Jeffrey Shallit,et al.  Automatic Sequences: Theory, Applications, Generalizations , 2003 .

[95]  Christian Mauduit.  Substitutions et ensembles normaux , 1989 .

[96]  Patrick Morton,et al.  Digit patterns and transcendental numbers , 1991, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[97]  E. Gilbert.  Gray codes and paths on the N-cube , 1958 .

[98]  J. Shallit,et al.  Automatic Sequences: Contents , 2003 .

[99]  N. Sloane.  A Handbook Of Integer Sequences , 1973 .