Geodesic
Beatty Sequences, Continued Fractions, and Certain Shift Operators
Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 473-482

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DOI

Let θ = θ(k) be the positive root of θ 2 + (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 [n θ] itself in this manner.
DOI : 10.4153/CMB-1976-071-6
Mots-clés : 10A35, 10A30, 10F20, Beatty sequences, complementary sequences of natural numbers, continued fractions, Fibonacci sequence, golden mean, greatest integer function, shift operators, Wythoff’s game 
Stolarsky, Kenneth B. Beatty Sequences, Continued Fractions, and Certain Shift Operators. Canadian mathematical bulletin, Tome 19 (1976) no. 4, pp. 473-482. doi: 10.4153/CMB-1976-071-6
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     title = {Beatty {Sequences,} {Continued} {Fractions,} and {Certain} {Shift} {Operators}},
     journal = {Canadian mathematical bulletin},
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     year = {1976},
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     doi = {10.4153/CMB-1976-071-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1976-071-6/}
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