The ring of k -regular sequences, II

Theoretical Computer Science

Volumn 307, Issue 1 SPEC., 2003, Pages 3-29

  Allouche, Jean Paul ^a   Shallit, Jeffrey ^b  

^a CNRS   (France)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Finite automata; k regular sequences

Indexed keywords

FINITE AUTOMATA; PROBLEM SOLVING; THEOREM PROVING;

TRANSITION FUNCTION;

COMPUTER SCIENCE;

EID: 0042235669     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(03)00090-2     Document Type: Conference Paper

References (51)

1. Allouche J.-P., Baake M., Cassaigne J., Damanik D. Palindrome complexity. Theoret. Comput. Sci. 292:2003;9-31.
2. Allouche J.-P., Shallit J.O. The ring of. k -regular sequences Theoret. Comput. Sci. 98:1992;163-197.
3. Allouche J.-P., Shallit J.O. The ubiquitous Prouhet-Thue-Morse sequence. Ding C., Helleseth T., Niederreiter H. Sequences and Their Applications, Proc. SETA '98. 1999;1-16 Springer, Berlin.
4. J.-P. Allouche, J.O. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003, to appear.
5. J.-P. Allouche, N. Rampersad, J. Shallit, On integer sequences whose first iterates are linear, submitted for publication, 2003.
6. Arkin J., Arney D.C., Dewald L.S., Ebel W.E. Jr. Families of recursive sequences. J. Recreational Math. 22:1990;85-94.
7. Babai L. On the length of subgroup chains in the symmetric group. Comm. Algebra. 14:1986;1729-1736.
8. E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning Ways for Your Mathematical Plays, Vol. 1, Academic Press, Toronto, 1982.
9. Cameron P.J., Solomon R., Turull A. Chains of subgroups in symmetric groups. J. Algebra. 127:1989;340-352.
10. Carlitz L. Single variable Bell polynomials. Collect. Math. 14:1962;13-25.
11. Carlitz L. A problem in partitions related to the Stirling numbers. Bull. Amer. Math. Soc. 70:1964;275-278.
12. Carlitz L. Some partition problems related to the Stirling numbers of the second kind. Acta Arith. 10:1965;409-422.
13. J. Cassaigne, Counting overlap-free binary words, in: P. Enjalbert, A. Finkel, K.W. Wagner (Eds.), STACS 93, Proc. 10th Symp. Theoretical Aspects of Computer Science, Vol. 665, Lecture Notes in Computer Science, Springer, Berlin, 1993, pp. 216-225.
14. Chang K.-N., Tsai S.-C. Exact solution of a minimal recurrence. Inform. Process. Lett. 75:2000;61-64.
15. Cobham A. On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory. 3:1969;186-192.
16. Cobham A. Uniform tag sequences. Math. Systems Theory. 6:1972;164-192.
17. Conway J.H. On Numbers and Games. 1976;Academic Press, New York.
18. P. Dumas, Récurrences Mahlériennes, Suites Automatiques, Études Asymptotiques, Thèse de Doctorat, Université Bordeaux I, 1993. INRIA Rapport 952, available from ftp://ftp.inria.fr/INRIA/publication/Theses/TU-0252/.
19. S. Eilenberg, Automata, Languages, and Machines, Vol. A, Academic Press, New York, 1974.
20. K.B. Ellul, J. Shallit, M.w. Wang, Regular expressions: new results and open problems, in: J. Dassow, M. Hoeberechts, H. Jürgensen, D. Wotschke (Eds.), Descriptional Complexity of Formal Systems (DCFS), Pre-Proceedings, 2002, pp. 17-34; Tech. Report No. 586, University of Western Ontario.
21. Graham R., Knuth D.E., Patashnik O. Concrete Mathematics. 1989;Addison-Wesley, Reading, MA.
22. R.K. Guy, Impartial games, in: R.K. Guy (Ed.), Combinatorial Games, Proc. Symp. Applied Mathematics, vol. 43, American Mathematical Society, Providence, RI, 1991, pp. 35-55.
23. Honsberger R. Mathematical Gems III. 1985;Mathematical Association of America, Washington, DC.
24. Kimberling C. Numeration systems and fractal sequences. Acta Arith. 73:1995;103-117.
25. Kuczma M.E. Problem 1922. Crux Math. 20:1994;74. (Solution in 21 (1995) 62-64).
26. E. Kullberg, Beyond infinity: on the infinity series - the DNA of hierarchical music, in: A. Beyer (Ed.), The Music of Per N ø rgård: Fourteen Interpretive Essays, Scolar Press, Aldershot, Hants, England, 1996, pp. 71-93.
27. Lam T.Y. The Algebraic Theory of Quadratic Forms. 1973;Benjamin, New York.
28. Lambek J., Moser L. On some two way classifications of integers. Canad. Math. Bull. 2:1959;85-89.
29. Mahler K. The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetic functions. Part Two. on the translation properties of a simple class of arithmetic functions J. Math. Phys. 6:1927;158-163.
30. Morton P., Mourant W. Paper folding, digit patterns, and groups of arithmetic fractals. Proc. London Math. Soc. 59:1989;253-293.
31. Morton P., Mourant W.J. Digit patterns and transcendental numbers. J. Austral. Math. Soc. Ser. A. 51:1991;216-236.
32. Mossé B. Reconnaissabilité des substitutions et complexité des suites automatiques. Bull. Soc. Math. France. 124:1996;329-346.
33. Patruno G. Solution to problem proposal 474. Crux Math. 6:1980;198.
34. Pfister A. Zur Darstellung von. -1 als Summe von Quadraten in einem Körper J. London Math. Soc. 40:1965;159-165.
35. Pfister A. Quadratische Formen in beliebigen Körpern. Invent. Math. 1:1966;116-132.
36. Porges A. A set of eight numbers. Amer. Math. Monthly. 52:1945;379-382.
37. Propp J. Problem proposal 474. Crux Math. 5:1979;229.
38. Roberts J. Lure of the Integers. 1992;Mathematical Association of America.
39. R. Rumely, Notes on van der Poorten's proof of the Hadamard quotient theorem: Part I, in: Séminaire de Théorie des Nombres, Paris, 1986-1987, Progress in Mathematics, Vol. 75, Birkhäuser, Boston, 1989, pp. 349-382.
40. R. Rumely, Notes on van der Poorten's proof of the Hadamard quotient theorem: Part II, in: Séminaire de Théorie des Nombres, Paris, 1986-1987, Progress in Mathematics, Vol. 75, Birkhäuser, Boston, 1989, pp. 383-409.
41. O. Salon, Propriétés arithmétiques des automates multidimensionnels, Thèse, Université Bordeaux I, 1989.
42. J. Shallit, The mathematics of Per N ø rgård's rhythmic infinity system, November 2002, submitted for publication.
43. Shapiro D.B. Products of sums of squares. Exposition. Math. 2:1984;235-261.
44. N.J.A. Sloane, The on-line encyclopedia of integer sequences, 2002. http://www.research.att.com/̃njas/sequences/.
45. Sloane N.J.A., Plouffe S. The Encyclopedia of Integer Sequences. 1995;Academic Press, New York.
46. R.G. Stanton, W.L. Kocay, P.H. Dirksen, Computation of a combinatorial function, in: C.St.J.A. Nash-Williams, J. Sheehan (Eds.), Proc. 5th British Combinatorial Conf. ( = Congressus Numerantium, Vol. 15), 1975, pp. 569-578.
47. Stewart C. Sums of functions of digits. Canad. J. Math. 12:1960;374-389.
48. Strauss N., Shallit J. Advanced problem 6625. Amer. Math. Monthly. 97:1990;252.
49. A. Weitzman, Transformation of parallel programs guided by micro-analysis, in: B. Salvy (Ed.), Algorithms Seminar, 1992-1993, Institut National de Recherche en Informatique et en Automatique, France, December 1993, pp. 155-159, Rapport de Recherche, No. 2130.
50. A.M. Yaglom, I.M. Yaglom, Challenging Mathematical Problems with Elementary Solutions, Vol. II. Holden-Day, San Francisco, CA, 1967 (Translated by J. McCawley).
51. Zagier D. Solution to advanced problem 6625. Amer. Math. Monthly. 99:1992;66-69.