Dulucq, S.; Gouyou-Beauchamps, D. Sur les facteurs des suites de Sturm. (On the factors of the Sturmian sequences.). (French) Zbl 0694.68048 Theor. Comput. Sci. 71, No. 3, 381-400 (1990). Summary: We study a construction which connects to each line with slope p/q (such that \(\gcd (p,q)=1\) and \(q\leq n)\) a word of length n over the alphabet \(\{\) 0,1\(\}\). We show that this construction yields the language of all the factors of the Sturmian sequences. We first obtain a functional equation whose solution is the generating function of this language, and then we give a conjecture relating this generating function to the Euler function. Cited in 1 ReviewCited in 37 Documents MSC: 68Q45 Formal languages and automata 05A15 Exact enumeration problems, generating functions Keywords:language; Sturmian sequences; generating function PDF BibTeX XML Cite \textit{S. Dulucq} and \textit{D. Gouyou-Beauchamps}, Theor. Comput. Sci. 71, No. 3, 381--400 (1990; Zbl 0694.68048) Full Text: DOI Online Encyclopedia of Integer Sequences: The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit. Number of words of length n in a certain language. Triangle of numbers a(i,j), i+j = n >= 2, giving number of words in a certain language with i 0’s, j 1’s, ending with 1. References: [1] Autebert, J. M.; Beququier, J.; Boasson, L.; Nivat, M., Quelques problèmes ouverts en théorie des langages algébriques, RAIRO Theoret. Comput. Sci., 13, 363-379 (1979) · Zbl 0434.68056 [2] Berstel, J., Transductions and Context-free Languages (1979), Teubner: Teubner Stuttgart · Zbl 0424.68040 [3] Berstel, J., Every iterated morphism yields a co-CFL, Inform. Process. Lett., 22, 7-9 (1986) · Zbl 0584.68082 [4] Christoffel, E. B., Observatio Arithmetica, Ann. Math., 6, 2, 148-152 (1875) · JFM 06.0113.02 [5] Cohen, H., Communication personelle (1987) [6] Coven, E. M.; Hedlund, G. A., Sequences with minimal block growth, Math. Systems Theory, 7, 138-153 (1983) · Zbl 0256.54028 [7] Flajolet, P., Analytic models and ambiguity of context-free languages, Theoret. Comput. Sci., 49, 283-309 (1987) · Zbl 0612.68069 [8] Grunbaum, B.; Shephard, G. C., Tilings and Patterns (1986), Freeman: Freeman San Francisco · Zbl 0601.05001 [9] Hedlund, G. A., Sturmian minimal sets, Amer. J. Math., 66, 605-620 (1944) · Zbl 0063.01982 [10] Hedlund, G. A.; Morse, M., Symbolic dynamics, Amer. J. Math., 60, 815-866 (1938) · JFM 64.0798.04 [11] Hedlund, G. A.; Morse, M., Symbolic dynamics, part II: Sturmian trajectories, Amer. J. Math., 62, 1-42 (1940) · JFM 66.0188.03 [13] Lunnon, W. F.; Pleasants, P. A.B., Quasicrystallographie Tilings, J. Math. Pure Appl., 66, 217-263 (1987) · Zbl 0626.52017 [14] Markoff, A. A., Sur une question de Jean Bernoulli, Math. Ann., 19, 27-36 (1882) · JFM 13.0313.01 [15] Rauzy, G., Suites à termes dans un alphabet fini, Séminaire de Théorie des Nombres de Bordeaux, 25.01-25.16 (1982-1983) [16] Rauzy, G., Mots infinis en arithmétique, dans: Automata on Infinite Words, (Ecole de Printemps d’Informatique Théorique. Ecole de Printemps d’Informatique Théorique, Lecture Notes in Computer Science, 192 (1985), Springer: Springer Berlin), 165-171 · Zbl 0613.10044 [17] Series, C., The geometry of Markoff numbers, Math. Intelligencer, 7, 20-29 (1985) · Zbl 0566.10024 [18] Smith, H. J.S., Note on continued fractions, Messenger Math., 6, 2, 1-14 (1876) [19] Stolarsky, K. B., Beatty sequences, continued fractions and certain shift operators, Canad. Math. Bull., 19, 473-482 (1976) · Zbl 0359.10028 [20] Venkov, B. A., (Elementary Number Theory (1970), Wolters-Noordhoff: Wolters-Noordhoff Groningen, The Netherlands), 67 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.