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A098457
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Farey Bisection Expansion of Sqrt[7].
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2
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1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| We define the Farey Bisection Expansion (FBE) of the nonnegative real number x to be the sequence {a(n)} of 0's and 1's determined as follows. Set na(0)=0, da(0)=1, nb(0)=1 and db(0)=0. For n=1, 2, 3,..., set num=na(n-1)+nb(n-1) and den=da(n-1)+db(n-1); if x<n/b, set a(n)=0, na(n)=na(n-1), da(n)=da(n-1), nb(n)=num, db(n)=den, else set a(n)=1, na(n)=num, da(n)=den, nb(n)=nb(n-1), db(n)=db(n-1). (The process is akin to that of locating the zero of a function by the bisection method, simply recording which successive subinterval, the left or the right, the zero lies at each refinement.) The FBE of Sqrt[7] is periodic with period 7. The RUNS transform of FBE(x) is the sequence of partial quotients of the continued fraction of x. As can be seen, RUNS(FBE(Sqrt[7]))={2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1,...}, which is A010121.
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LINKS
| N. J. A. Sloane, Stern-Brocot or Farey Tree
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FORMULA
| a(n)=1/147*{5*(n mod 7)+5*[(n+1) mod 7]-16*[(n+2) mod 7]+26*[(n+3) mod 7]-16*[(n+4) mod 7]+26*[(n+5) mod 7]+5*[(n+6) mod 7]} - Paolo P. Lava (paoloplava(AT)gmail.com), Nov 21 2006
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CROSSREFS
| Cf. A010121, A097853, A098458.
Sequence in context: A065251 A039982 A131372 * A137161 A077050 A128432
Adjacent sequences: A098454 A098455 A098456 * A098458 A098459 A098460
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KEYWORD
| nonn
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AUTHOR
| John W. Layman (layman(AT)math.vt.edu), Sep 08 2004
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