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Beatty Sequences, Continued Fractions, and Certain Shift Operators

Published online by Cambridge University Press:  20 November 2018

Kenneth B. Stolarsky
Kenneth B. Stolarsky*
Affiliation:
Department of Mathematics, University of ColoradoBoulder, Colorado80302
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Abstract

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Let θ = θ(k) be the positive root of θ2 + (k-2)θ-k = 0. Let f(n) = [(n + l)θ]-[nθ] for positive integers n, where [x] denotes the greatest integer in x. Then the elements of the infinite sequence (f(l), f(2), f(3),…) can be rapidly generated from the finite sequence (f(l), f(2),…,f(k)) by means of certain shift operators. For k = 1 we can generate (the characteristic function of) the sequence [nθ] itself in this manner.

Keywords

10A3510A3010F20Beatty sequencescomplementary sequences of natural numberscontinued fractionsFibonacci sequencegolden meangreatest integer functionshift operatorsWythoff’s game
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 4 , 01 December 1976 , pp. 473 - 482
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Ahrens, W., Mathematische Unterhaltungen und Spiele, vol. 1, 2nd ed., Teubner, Leipzig, 1910 (pp. 84–88).Google Scholar
2. Angel, M., Partitions of the natural numbers, Canad. Math. Bull. 7 (1964), 219–236. MR 28 #5030.Google Scholar
3. Ball, W. W. R., Mathematical Recreations and Essays, revised by H. S. M. Coxeter, MacMillan, New York, 1939 (pp. 36–40).Google Scholar
4. Bang, Th., On the sequence [nα], n = 1, 2,…, Math. Scand. 5 (1957), 69–76. MR 19 (1958), p. 1159.Google Scholar
5. Beatty, S., Problem 3173, Amer. Math. Monthly 33 (1926), 159. Solutions, ibid., 34 (1927), 159.Google Scholar
6. Bernoulli, J. III, Sur une nouvelle espèce de calcul, Recueil pour les astronomes, vols. 1, 2, Berlin, 1772.Google Scholar
7. Bricard, R., Sur un problème relatif aux nombres incommensurables, Nouv. Ann. de Math. (6) 1 (1926), 100–103.Google Scholar
8. Carlitz, L., Hoggatt, V. E. Jr, and Scoville, R., Fibonacci representations, Fib. Quart. 10 (1972), 1–28.Google Scholar
9. Carlitz, L., Hoggatt, V. E. Jr, and Scoville, R., Lucas representations, Fib. Quart. 10 (1972), 29–42, 70, 112.Google Scholar
10. Carlitz, L., Hoggatt, V. E. Jr, and Scoville, R., Fibonacci representations of higher order, Fib. Quart. 10 (1972), 71–80.Google Scholar
11. Carlitz, L., Hoggatt, V. E. Jr, and Scoville, R., Fibonacci representations of higher order II, Fib. Quart. 10 (1972), 71–80.Google Scholar
12. Carlitz, L., Hoggatt, V. E. Jr, and Scoville, R., Pellian representations, Fib. Quart. 10 (1972), 449–488; Addendum, ibid., 527–530. MR 47 #8420.Google Scholar
13. Carlitz, L., Scoville, R., and Vaughan, T., Some arithmetic functions related to Fibonacci numbers, Fib. Quart. 11 (1973), 337–386. MR 48 #10969.Google Scholar
14. Cohn, H., Representation of Markoff’s binary quadratic forms by geodesies on a perforated torus, Acta Arith. 18 (1971), 125–136. MR 44 #5277.Google Scholar
15. Cohn, H., Markoff forms and primitive words, Math. Ann. 196 (1972), 8–22. MR 45 #6899.Google Scholar
16. Cohn, H., Some direct limits of primitive homotopy words and of Markoff geodesies, esp. pp. 97–98, in Discontinuous Groups and Riemann Surfaces, L. Greenberg éd., Ann. of Math. Studies 79, Princeton University Press, Princeton, 1974.Google Scholar
17. Cohn, H., Problem E2544, Amer. Math. Monthly 82 (1975), 660. Solution, ibid., 83 (1976), 745–747.Google Scholar
18. Connell, I. G., A generalization of Wythoff’s game, Canad. Math. Bull. 2 (1959), 181–190.Google Scholar
19. Connell, I. G., Some properties of Beatty sequences I, Canad. Math. Bull. 2 (1959), 190–197. MR #7805.Google Scholar
20. Connell, I. G., Some properties of Beatty sequences II, Canad. Math. Bull. 3 (1960), 17–22. MR 22 #1558.Google Scholar
21. Coxeter, H. S. M., The golden section, phyllotaxis and Wythoff’s game, Scripta Math. 19 (1953), 135–143. MR 15 (1954), p. 246.Google Scholar
22. Domoryad, A. P., Mathematical Games and Diversions (Russian), Gosudarstv. Izdat. Fiz- Mat. Lit., Moscow, 1961 (pp. 59–67). MR 23 #A761.Google Scholar
23. Fan, K., Problem 4399, Amer. Math. Monthly 57 (1950), 343. Solutions, ibid., 59 (1952), 48–49.Google Scholar
24. Faucheux, Sur une question concernant des suites de nombres incommensurables, Nouv. Ann. de Math. (6) 1 (1926), 237–239.Google Scholar
25. Feaenkel, A. S., The bracket function and complementary sets of integers, Can. J. Math. 21 (1969), 6–27. MR 38 #3214.Google Scholar
26. Feaenkel, A. S., Levitt, J., and Shimshoni, M., Characterization of the set of values f(n) = [nα], n = 1, 2,…, Discrete Math. 2 (1972), 335–345. MR 46 #1743.Google Scholar
27. Feaenkel, A. S., Complementing and exactly covering sequences, J. Comb. Theory Ser. A14 (1973), 8–20. MR 4£#8875.Google Scholar
28. Feaenkel, A. S., A characterization of exactly covering congruences, Discrete Math. 4 (1973), 359–366. MR 47 #4906.Google Scholar
29. Feaenkel, A. S. and Borosh, I., A generalization of Wythoff’s game, J. Comb. Theory 15 (1973), 175–191.Google Scholar
30. Fraenkel, A., Further characterizations and properties of exactly covering congruences, Discrete Math. 12 (1975), 93–100.Google Scholar
31. Fraenkel, A., Complementary systems of integers, in preparation.Google Scholar
32. Gilbert, E. N., Functions which represent all integers, Amer. Math. Monthly 70 (1963), 736–738; Acknowledgement, 1082.Google Scholar
33. Gould, H. W., Problem HA, Fib. Quart. 1 (1963), 46.Google Scholar
34. Gilbert, E. N., Generalization of a bracket formula of L. Moser, Canad. Math. Bull. 6 (1963), 275–277; Editor’s comment, 277–278.Google Scholar
35. Gilbert, E. N., Non-Fibonacci numbers, Fib. Quart. 3 (1965), 177–183. MR 35 #118.Google Scholar
36. Graham, R. L., On a theorem of Uspensky, Amer. Math. Monthly 70 (1963), 407–409. MR 26 #6062.Google Scholar
37. Graham, R. L., and Pollak, H. O., Note on a nonlinear recurrence related to , Math. Mag. 43 (1970), 143–145. MR 42 #180.Google Scholar
38. Graham, R. L., and Pollak, H. O., Covering the positive integers by disjoint sets of the form ﹛[nα + β]:n = 1,2,…﹜, J. Comb. Theory Ser. A15 (1973), 354–358.Google Scholar
39. Grossman, H. D., A set containing all integers, Amer. Math. Monthly 69 (1962), 532–533.Google Scholar
40. Hijikata, H., On a game of Wythoff (Japanese), Sûgaku 11 (1959/60), 220–222. MR 25 #42.Google Scholar
41. Holladay, J. C., Cartesian products of termination games, Contributions to the theory of games, Vol. 3, Princeton University Press, New Jersey, 1957. MR 20 #2236.Google Scholar
42. Holladay, J. C., Some convergent recursive sequences, homeomorphic identities, and inductively defined complementary sequences, Fib. Quart. 4 (1966), 1–36. MR 36 #5063.Google Scholar
43. Holladay, J. C., Some generalizations of Wythoff’s game and other related games, Math. Mag. 41 (1968), 7–13.Google Scholar
44. Honsberger, R., Ingenuity in Mathematics, Random House, New York, 1970 (pp. 93–110).Google Scholar
45. Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, John Wiley and Sons, New York, 1974 (pp. 305–318).Google Scholar
46. Lambek, J. and Moser, L., Inverse and complementary sequences of natural numbers, Amer. Math. Monthly 61 (1954), 454–458. MR 16 (1955), p. 17.Google Scholar
47. Leveque, W. J., editor, Reviews in Number Theory vol. 1, American Mathematical Society, Providence, Rhode Island, 1974 (pp. 331–333). MR #2040.Google Scholar
48. Markoff, A. A., Sur une question de Jean Bernoulli, Math. Ann. 19 (1882), 27–36.Google Scholar
49. Markoff, A. A., Collected Works (Russian), Izdat. Akad. Nauk. SSSR, Leningrad, 1951 (pp. 13, 15–16).Google Scholar
50. Miller, J. C. P., The sum of the integral parts in an arithmetical progression, Math. Gazette 36 (1952), 234–243. MR 14 (1953), p. 536.Google Scholar
51. Newman, D. J., Problem 5252, Amer. Math. Monthly 71 (1964), 1138. Solutions, ibid., 72 (1965), 1144–1145.Google Scholar
52. Niven, I., Diophantine Approximations, Interscience Tracts in Pure and Applied Mathematics #14, Interscience Publishers, John Wiley and Sons, New York, c. 1963 (pp. 34–45). MR 26 #6120.Google Scholar
53. Niven, I. and Zuckerman, H., An Introduction to the Theory of Numbers, John Wiley and Sons, New York, 2nd éd., 1966. MR 22#5605.Google Scholar
54. O’Beirne, T. H., Puzzles and Paradoxes, Oxford University Press, London, 1965 (pp. 130–138).Google Scholar
55. Skolem, Th., On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957), 57–68. MR 19 (1958), p. 1159.Google Scholar
56. Skolem, Th., Über einige Eigenschaften der Zahlenmengen [αn + β] bei irrationalem a mit einleitenden Bererkungen uber einige kombinatorische Problème, Norske Vid. Selsk. Forh. (Trondheim) 30 (1957), 118–125. MR 19 (1958), pp. 1159–1160.Google Scholar
57. Skolem, Th., Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958), 273–280. MR 19 (1958), p. 1159.Google Scholar
58. Sloane, N. J. A., A Handbook of Integer Sequences, Academic Press, New York, 1973 (pp. 29–30).Google Scholar
59. Stewart, B. M., Theory of Numbers, 2nd éd., MacMillan, New York, 1964 (pp. 76–77). MR 37 #6232.Google Scholar
60. Sprague, R., Ein Satz über Teilfolgen der Reihe der natürlichen Zahlen, Math. Ann. 115 (1938), 153–156.Google Scholar
61. Uspensky, J. V., On a problem arising out of the theory of a certain game, Amer. Math. Monthly 34 (1927), 516–521.Google Scholar
62. Uspensky, J. V., and Heaslet, M. A., Elementary Number Theory, McGraw-Hill, New York, 1939 (p. 98). MR 1 (1940), p. 38.Google Scholar
63. Uspensky, J. V., and Heaslet, M. A., On a problem of John Bernoulli. I, II, III, Revista Union Mat. Argentina 11 (1946), 141–154; 164–183; 239–255. MR 8 (1947), pp. 5, 6, 443.Google Scholar
64. Uspensky, J. V., and Heaslet, M. A., On a problem of John Bernoulli, IV, Revista Union Mat. Argentina 12 (1946), 10–19. MR 8 (1947), p. 443.Google Scholar
65. Venkov, B. A., Elementary Number Theory, Translated and edited by Helen Alderson, Wolters-Noordhoff, Groningen, 1970 (pp. 65–68). MR 42 #178.Google Scholar
66. Vinogradov, I. M., Elements of Number Theory, translated from the fifth revised edition by Kravetz, S., Dover, 1954 (p. 29). MR 15 (1954), p. 933.Google Scholar
67. Wythoff, W. A., A modification of the game of Nim, Nieuw. Archief voor Viskunde (2), 7 (1907), 199–202.Google Scholar
68. Yaglom, A. M. and Yaglom, I. M., Challenging Mathematical Problems with Elementary Solutions, translated by McCawley, J. Jr, revised and edited by Gordon, B., vol. 2, Holden-Day, San Francisco, 1967 (pp. 20, 105–112). MR 29 #2193.Google Scholar