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A194368
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Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - <k*r>) = 0, where r=sqrt(2) and < > denotes fractional part.
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68
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2, 4, 12, 14, 16, 24, 26, 28, 70, 72, 74, 82, 84, 86, 94, 96, 98, 140, 142, 144, 152, 154, 156, 164, 166, 168, 408, 410, 412, 420, 422, 424, 432, 434, 436, 478, 480, 482, 490, 492, 494, 502, 504, 506, 548, 550, 552, 560, 562, 564, 572, 574, 576, 816, 818
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OFFSET
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1,1
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COMMENTS
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Suppose that r and c are real numbers, 0 < c < 1, and
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s(m) = Sum_{k=1..m} (<c+k*r> - <k*r>)
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where < > denotes fractional part. The inequalities s(m) < 0, s(m) = 0, s(m) > 0 yield up to three sequences that partition the set of positive integers, as in the examples cited below. Of particular interest are choices of r and c for which s(m) >= 0 for every m >= 1.
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Note that s(m) = m*c - Sum_{k=1..m} floor(c + <k*r>). This shows that if c is a rational number p/q, then the range of s(m) is a set of rational numbers having denominator q. In this case, it is easy to prove that if s(m)=0, then m is an integer multiple of q, yielding a sequence of quotients denoted by [[m/q>]] in the following list:
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r..........p/q....s(m)<0....s(m)=0....[[m/q]]...s(m)>0
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Next, suppose that r and c are chosen so that s(m)=0 for all m. Then the sets X={m : s(m)<0} and Y={m : s(m)>0} represent a pair of "generalized Beatty sequences" in this sense: if c=1/<r>, the sets X and Y represent the Beatty sequences of 1/<r> and 1<-r>. Examples:
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r..........c.........X.........Y......
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REFERENCES
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Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963.
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LINKS
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Ronald L. Graham, Shen Lin, Chio-Shih Lin, Spectra of numbers, Math. Mag. 51 (1978), 174-176.
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MATHEMATICA
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r = Sqrt[2]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]] (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]] (* A194368 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]] (* A194370 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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