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Search: id:A079934
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| A079934 |
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Greedy frac multiples of sqrt(2): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(2). |
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+0
8
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| 1, 3, 5, 10, 17, 29, 46, 99, 169, 268, 577, 985, 1562, 3363, 5741, 9104, 19601, 33461, 53062, 114243, 195025, 309268, 665857, 1136689, 1802546, 3880899, 6625109, 10506008, 22619537, 38613965, 61233502, 131836323, 225058681, 356895004
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The n-th greedy frac multiple of x is the smallest integer that does not cause sum(k=1..n,frac(a(k)*x)) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.
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FORMULA
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For n>0, a(3n) = A000129(2n+1), a(3n+2) = a(3n) + A000129(2n+2) and a(3n+4) = a(3n+2) + a(3n+3). Also a(3n) = ceil((3+2*sqrt(2))^n*(2+sqrt(2))/4). a(3n+2)/a(3n+1) -> 1/sqrt(2); a(3n+1)/a(3n) -> 3-sqrt(2); a(3n)/a(3n-1) -> (8+5*sqrt(2))/7.
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EXAMPLE
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a(4) = 10 since frac(1x) + frac(3x) + frac(5x) + frac(10x) < 1, while frac(1x) + frac(3x) + frac(5x) + frac(k*x) > 1 for all k>5 and k<10.
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CROSSREFS
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Cf. A000129 (Pell numbers), A078343, A079935, A079936.
Sequence in context: A000990 A129361 A062773 this_sequence A005403 A018072 A090170
Adjacent sequences: A079931 A079932 A079933 this_sequence A079935 A079936 A079937
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KEYWORD
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nonn
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr) and Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2003
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