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Search: id:A105960
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| A105960 |
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Smallest integer q >= 1 such that difference between q*sqrt(2) and the nearest integer is <= 1/n. |
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3
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| 1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29
(list; graph; listen)
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OFFSET
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2,2
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COMMENT
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Theorem 1 in Cassels says given real numbers x and Q>1, there is an integer q such that 0 < q < Q and the difference between qx and the nearest integer is <= 1/Q. This sequence arises from taking x = sqrt(2) and Q = n = 2,3,4,...
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REFERENCES
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J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge, 1957.
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MAPLE
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Digits:=200; M1:=200; th:=x->abs(x-round(x)); f:=proc(x) local Q, q, t1, x1; t1:=[]; for Q from 2 to M1 do x1:=evalf(1/Q); q:=1; while th(q*x) > x1 do q:=q+1; od; t1:=[op(t1), q]; od; t1; end; f(evalf(sqrt(2)));
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CROSSREFS
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Cf. Pell numbers A000129; A108688, A108689.
Adjacent sequences: A105957 A105958 A105959 this_sequence A105961 A105962 A105963
Sequence in context: A103286 A058704 A098101 this_sequence A081290 A123081 A020917
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KEYWORD
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nonn
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AUTHOR
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njas, Jun 18 2005
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