Beatty Sequences, Continued Fractions, and Certain Shift Operators

Venkov, Elementary Number Theory, 1970, 65

Uspensky, 1940, Elementary Number Theory, 1939

10.1007/BF01448934

10.7146/math.scand.a-10490

O’Beirne, Oxford University Press, London,, 1965, 130

Niven, 1966, An Introduction to the Theory of Numbers

Newman, 1964, Problem 5252, Amer. Math. Monthly, 71, 1144

10.2307/3608198

10.1007/BF01447292

10.1080/00029890.1954.11988496

Markoff, Collected Works, Izdat. Akad. Nauk. SSSR, Leningrad, 1951, 15

Kuipers, Uniform Distribution of Sequences, 1974, 305

Honsberger, Ingenuity in Mathematics, 1970, 93

10.1080/0025570X.1968.11975824

Holladay, 1957, Cartesian products of termination games, 3

Hijikata, On a game of Wythoff, 11, 220

10.2307/2311195

10.2307/2311859

Yaglom, 1967, Challenging Mathematical Problems with Elementary Solutions, 105

Vinogradov, 1954, Elements of Number Theory, 1954, 933

Gilbert, 1963, Generalization of a bracket formula of L. Moser, Canad. Math. Bull., 6, 275, 10.4153/CMB-1963-025-2

Gould, 1963, Problem HA, Fib. Quart., 1

Fraenkel, Complementary systems of integers

10.1016/0012-365X(75)90098-9

10.1016/0012-365X(73)90170-2

1926, Sur une question concernant des suites de nombres incommensurables, Nouv. Ann. de Math. (6), 1, 237

Fan, 1952, Problem 4399, Amer. Math, 59, 48

Domoryad, Mathematical Games and Diversions, Izdat. Fiz- Mat. Lit., 1961, 59

Coxeter, 1953, The golden section, phyllotaxis and Wythoff’s game, Scripta Math., 19, 135

10.4153/CMB-1960-004-2

10.4153/CMB-1959-025-0

10.4153/CMB-1959-024-3

10.2307/2312258

Cohn, 1975, Problem E2544, Amer. Math. Monthly, 82, 745

Cohn, 1974, Some direct limits of primitive homotopy words and of Markoff geodesies, 97

Niven, 1963, Diophantine Approximations, Interscience Tracts in Pure and Applied Mathematics #14, Interscience Publishers, John Wiley and Sons, New York, c., 34

10.4064/aa-18-1-125-136

Skolem, 1957, Über einige Eigenschaften der Zahlenmengen [αn + β] bei irrationalem a mit einleitenden Bererkungen uber einige kombinatorische Problème, Norske Vid. Selsk. Forh. (Trondheim), 30, 118

Carlitz, 1973, Some arithmetic functions related to Fibonacci numbers, Fib. Quart., 11, 337

Carlitz, 1972, Pellian representations, Fib. Quart., 10, 449

10.1016/0097-3165(73)90059-9

10.4153/CJM-1969-002-7

Gilbert, 1965, Non-Fibonacci numbers, Fib. Quart., 3, 177

10.1016/S0097-3165(73)80005-6

Sloane, Academic Press, New York,, 1973, 29

10.1016/0097-3165(73)90084-8

Beatty, 1926, Problem, Amer. Math. Monthly, 33

Carlitz, 1972, Fibonacci representations of higher order, Fib. Quart., 10, 71

Leveque, 1974, Reviews in Number Theory, 331

Stewart, Theory of Numbers, 1964, 76

Bricard, 1926, Sur un problème relatif aux nombres incommensurables, Nouv. Ann. de Math. (6), 1, 100

Uspensky, 1946, On a problem of John Bernoulli, IV, Revista Union Mat. Argentina, 10

Graham, 1970, Note on a nonlinear recurrence related to, Math. Mag., 43, 143

Ball, Mathematical Recreations and Essays, revised by H. S. M. Coxeter, MacMillan, New York,, 1939, 36

10.1007/BF01419427

Bernoulli, 1772, Sur une nouvelle espèce de calcul, Recueil pour les astronomes

10.1080/00029890.1927.11986764

10.1016/0012-365X(72)90012-X

10.7146/math.scand.a-10551

10.7146/math.scand.a-10491

Ahrens, 1910, Mathematische Unterhaltungen und Spiele, 84

Carlitz, 1972, Fibonacci representations of higher order II, Fib. Quart., 10, 71

Uspensky, 1947, On a problem of John Bernoulli. I, II, III, Revista Union Mat. Argentina 11 (1946), 141–154; 164–183; 239–255. MR, 8

Wythoff, 1907, A modification of the game of Nim, Nieuw. Archief voor Viskunde (2),, 7, 199

Holladay, 1966, Some convergent recursive sequences, homeomorphic identities, and inductively defined complementary sequences, Fib. Quart., 4, 1

10.4153/CMB-1964-020-1

Carlitz, 1972, Fibonacci representations, Fib. Quart., 10, 1

Carlitz, 1972, Lucas representations, Fib. Quart., 10, 29