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Search: id:A079936
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| A079936 |
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Greedy frac multiples of sqrt(5): a(1)=1, sum(n>0,frac(a(n)*x))=1 at x=sqrt(5). |
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+0
3
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| 1, 2, 5, 13, 17, 34, 305, 610, 1597, 4181, 5473, 10946, 98209, 196418, 514229, 1346269, 1762289, 3524578, 31622993, 63245986, 165580141, 433494437, 567451585, 1134903170, 10182505537, 20365011074, 53316291173, 139583862445
(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(6n+1)=A001076(4n+1); a(6n+2)=2a(6n+1); a(6n+3)=A001076(4n+1)+A001076(4n+2); a(6n+4)=A001076(4n+3)-A001076(4n+2); a(6n+5)=A001076(4n+3); a(6n+6)=2a(6n+5). Asymptotics: a(6n) -> 2*sqrt(5)*(tau)^(12n-3); a(6n+2)/a(6n+1) -> (tau)^2; a(6n+3)/a(6n+2) -> (tau)^2; a(6n+4)/a(6n+3) -> (tau)^2/2; a(6n+6)/a(6n+5) -> (tau)^6/2; where tau = (1+sqrt(5))/2.
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EXAMPLE
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a(4) = 13 since frac(1x) + frac(2x) + frac(5x) + frac(13x) < 1, while frac(1x) + frac(2x) + frac(5x) + frac(k*x) > 1 for all k>5 and k<13.
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CROSSREFS
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Cf. A001076 (denominators of convergents to sqrt(5)), A079934, A079935, A079937.
Sequence in context: A074856 A087952 A124255 this_sequence A102854 A156009 A164620
Adjacent sequences: A079933 A079934 A079935 this_sequence A079937 A079938 A079939
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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 21 2003
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